clapton davis fic where hes just like, super flirty and its really cute and the reader is oblivious to this but eventually clapton is like "damn it why cant you get the hint" so he opens up to the reader?&;&:& tysmm

━━ UNSUBTLE SUBTILITY

'୧ ‧₊ pairing: clapton davis x reader warnings: swearing, brief depictions of blood word count: 2500+ ⋆ ✩‧₊

The presence of Spring in Grizzly Lake brought a lot of things; including sporadic bursts of heaven-yellow sunlight, greenery spiraled across branches of previously barren tree skeletons, and, most importantly for students of Grizzly Lake High School, the promise of the Spring Fling Formal that was set to occur in the midst of May. 

For Clapton, this prom meant one thing; achieving his goal that’s been looming over him since freshman year — ask you out. Theoretically it’s a simple process, but if it was truly as easy as it sounds it would have occurred the very moment his eyes landed on your figure that first day in beginner spanish. 

You were the embodiment of perfection, punctuated through your gleaming smile that enraptured anyone in a ten mile radius, and the way the sun seemed to spread across the expanse of your cheeks, soaking you in the rays of heaven itself. Clapton was about ready to propose that day, and he didn’t even know your name. 

Now, roughly two years later, he was still amidst the same dilemma, the one in which he actually had to do the asking-out part. He was sure by now you would have picked up on his inherently obvious attempts to entice you, but you remained oblivious, so he decided he’d have to fully commit if he wanted to capture your attention. The art of unsubtle subtility, if you will. 

And so, forty three minutes into the depths of an agonizingly dull pre-calculus lesson, he confidently taps your shoulder with a fractionally tense hand, and indulges the tug on his heartstrings when you turn around, framed by the delicate glow of mid-morning spring that he adores so much. 

“Something wrong, Clapton?” Your voice cleaves through the classroom ambience of idle chatter and textbook pages being flipped. He flashes a boyish smile in hopes to flutter your heart in the same way you flutter his. 

“Do you get any of these questions?” 

“Yeah, they’re not too bad,” you reply, offering an ephemeral that renders his throat tight. 

He glanced down momentarily at his worksheet, adorned in scrawls and scribbles, yet lacking a single legible answer. His vision trains up back to you though, as it always does. He thought you’d easily detect the unspoken question for your help, but you remained stationary in your seat, as if waiting for him to say it. He couldn’t tell if you were genuinely that heedless, or if you were toying with him. Cat and mouse. 

“Seriously? When did they even teach us all this?”

You shrug mindlessly, and a lock of hair shifts from its position on your shoulder. He’d give anything to rope his fingers through it. “A while back. Why, you need some help?” 

Yes. He’d like your help, your compassion, your hand in marriage…

“Wanna walk me through it?” He tosses you a hopeful expression, and you answer back with a simple nod, sliding your chair along the cheap linoleum floor with a scrape, until the pair of you are sharing his desk, impossibly close. 

Your velvet voice is stringing sentences right down the expanse of his spine, though your attempts to help him understand logarithmic differentiation were ultimately futile— how was he supposed to concentrate on anything when he could feel your words blooming on his skin? See every freckle and divot etched into your face? He could taste his own heartbeat as it melded against his throat.

“So, this helps to avoid complications like the product rule and the quotient rule when— Clapton?”

He cocks his head up, trying to ignore the swell in his stomach when he hears the way his name sounds braided between your sentences, it suits your voice so well.

“Yeah? What’s up?” 

“Are you even listening?”  

Shit, no he absolutely wasn’t. How could he? Your proximity allowed him to see you. Like, properly see you. 

“Yeah. Totally. Logaramic thingyation,” he murmurs with overt certainty, and a puppylike grin. 

You snicker. “Couldn’t even get the name right?” 

He’s internally collapsing, though he manages to force some words out of his struggling brain. 

“Hard to think when you’re here.” He doesn’t dare sever the eye contact between you, hoping to hone the tension as long as possible, until he shatters you. His lopsided grin shrinks in a moment of brevity; you’re so close and he can smell you and your very essence. He’s sure that his ulterior motive is conveyed, through the way his eyes explore the breadth of your figure, never leaving, never faltering— yet to his pure irritation, all he gets is a blank expression and a confused chuckle. 

“Why is that?” You ask, and he wants to grab you by your shoulders and shake you. Are you really that dense? Your face is about as expressive as a rock, and you seem not even partially affected by the flirty wink he sent your way moments prior. 

“You’re kidding, right? Come on.” He fires back, raising a brow with a daring smirk. He wants you to inquire. You don’t. He realizes that trying to get you to take a fucking hint was about as impossible as teaching him calculus. 

You force out an awkward laugh that makes his skin crawl with defeat, but he doesn’t back down. “Come on what?” 

He refrains from the urge to say “me”, and instead huffs a sharp exhale through his nose. He’s moments away from spouting some lame compliment when the shrill cry of the bell interrupts his train of thought, and a tide of students eject eagerly from their seats and spill out into the corridor for lunch. 

Your friend approaches the desk with a quirked brow, reaching for your arm and mumbling something into your ear that’s intelligible to Clapton, tugging on you to try and steer you away from the classroom. And from him. You nod in response to her comment, before momentarily glancing back over to Clapton.

“I gotta go, Clapton. See you soon though, see you in History!” You send him a parting wave with a gentle flick of your wrist, before turning off and disappearing down the long stretch of corridor beside the classroom. His eyes follow you for as long as possible before your figure is consumed by the wandering horde of students, and he lets a grumbly sigh escape his parted lips before he packs up his belongings. This was going to be harder than he anticipated. 

*:･.・゜゜・

Clapton’s second attempt at alluring you resulted in more or less the same outcome. He’d entered the cafeteria, instantly bathed in the overwhelming odor of lysol and lard. His prior plan was to grab a doctor pepper, maybe a sandwich, and head over to his typical table to talk a painfully uninterested Sander’s ear off about you, but he scrapped it upon spotting you waiting in the cafeteria line, immediately changing course and veering over in hopes of a successful conversation.

He cuts in front of an unsuspecting freshman, ignores the irritated “What’s your deal man?”, and ‘accidentally’ brushes up to you until your bodies knock, and you spin around in confusion. 

Your face mildly relaxes in recognition, and he takes this as progress.

 “Hey. Getting lunch?”

“What else would I be doing?” You ask. Swing and a miss. 

He clears his throat a fraction, not allowing this to throw him off his game. 

“I dunno, maybe you just really like standing in lines,” he teases, and you laugh back. 

“Especially if the line is for overpriced cafeteria food,” you add with a grin.

The pair of you share a laugh, and Clapton marvels at the fact that you can look so irresistible even in the harsh fluorescence of the cafeteria’s artificial lighting. The pair of you fall into a partially awkward silence, and he follows your line of vision, watching as you observe some students hanging a hand painted banner advertising prom for the entirety of the cafeteria to see. ‘Spring Fling Formal, get your tickets now!’ glistens in white gold lettering. He prays he can take the banner up on that offer. 

“Are you doing anything for it?” A bit of a jump from the casual conversation, but he was itching to entice you and couldn’t risk missing his chance. 

“Hm? For what?” His lips twitch into a gradually familiar downwards smile. “Prom,” he says, gesturing at the banner, obnoxiously pink in hue and decorated with scatterings of hastily painted daisies. 

“Oh. Maybe— I’m not sure, it’s kinda ages away.” Yup. An impossibly distant period of two weeks. Clapton’s jaw ticks uncomfortably at the prospect of the narrowing window of time. He can’t afford to screw this up.

“Right. Sure. Are you… interested in anyone in particular though?” He probes, hoping that you notice the searing spark of desperation that lingers in the loop of his irises.

“Eh. Not really. Are you?”

His ego suffers a blow at your total ignorance to his pining. He’s on the brink of combustion; unable to endure the cosmic irony of having you so close yet so far. He pictures you for the umpteenth time, glittering in a dress that matched your eyes and his tie. A slow dance to a Sting song, his eager hands situated either side of your waist. You’d stare up at him with a dazzled guise, illuminated by the scintillation of indigo disco lights, and his tongue would delve into yours as he soaked up the saccharine flavor of the fruit punch lingering on your lips. 

“Yeah.” He states bluntly, staring at you as if you hung each and every star. “Yeah, I’m interested in someone.” 

You raise a brow. “Oh yeah? Who?”

He clears his throat. “Someone special. Someone super special.”

“You should ask them!” “Easier said than done,” he chuckles humorlessly. 

Your lips part as you go to investigate further, but are interrupted by the scowl of the lunch lady barking at you for your order. He notes it, mac and cheese plus a diet sprite— you’re handed it moments later, and your vision is torn from him and towards your small circle of friends seated across the cafeteria, who are waving you down. You’re gonna leave again? 

“I better go sit down, but, uh, you should definitely ask that person to prom. Be upfront and everything. Y’know, you only live once, and all that, right?” 

He swears he’s going to implode at the unbridled irony of this entire situation. Be upfront. He’s been upfront! 

“You know it,” he quips weakly as you slink away. 

He’s been showering you in signals for months, and you’d always abandon them, his attempts for your acknowledgement left festering as sour memories in his head, things that made him roll over with shame in bed at night, and all for what?

He brainlessly orders his doctor pepper with a monotone grumble, feeling the frigid prick of the can’s condensation gather in his palm as he wonders what the hell it’s gonna take for you to take a damn hint. 

*:･.・゜゜・

After yet another failed interaction, Clapton had spent the span of the rest of the week stripping his words to the marrow. Every conversation he indulged in with you involved his inner thoughts spouted in their rawest form— cocky compliments, lingering touches, looks of intense pining and yet somehow you continued to miss them. Every. Last. One. 

He was nearing his wits end, teetering on the cliff of insanity and seconds away from taking the plunge. Maybe he was the one who needed to take a hint. Maybe you were trying to tell him that you weren’t interested and he wasn’t giving it up. It was a sickening notion, one that thrashes wildly in his stomach. He didn’t know much, but he did know that he’d never be satisfied until he knew your stance on him for certain.  

He was just gonna say it. 

In hindsight, it wasn’t Clapton’s smartest move to deliver the question in the midst of a dodgeball game, but his thoughts were warped and he decided now was as good as ever. His voice was barely even audible beside you over the screech of tennis sneakers scraping the gym floor and the continuous sound of rubber balls coming into contact with student flesh. 

“Hey!” He exclaims. 

“Hey?” You say back, turning to him momentarily. Yet again, he wonders how you do it. Hair blown back effortlessly, skin glistening with a fragile sheen of moisture that is hardly off-putting, if doing something it aids to soften your otherworldly glow. Meanwhile, he was panting like an old dog, hair matted to his forehead in sodden chunks beneath his obnoxious sweatband. 

“I needa ask you something!” It’s sink or swim. His teeth graze the inside of his cheek for a moment, his gaze varying between you and the opposing court, to prevent a dodgeball to the head. 

“Yeah?” Sink or swim sink or swim sink or swim. “What’s up?” He melts at the sight of your semi-breathless smile.

“Are you still dateless? Like, to prom?”

Your forehead creases, and you return the sideways glance. “Um, yeah. Why?”

With a delayed exhale that rings heavy in the pits of his lungs, he turns his entire body to face you, which in turn makes you face him as well. 

“Look, I’ve been trying to say this for months. Well, not months. Maybe weeks. Whatever– point is, it’s been a while. Like seriously, a long fucking time. And I swear I’ve been so obvious, but clearly not obvious enough because you’re still, like, totally unaware or whatever. But, like, basically, I was wondering— I’ve been wondering if—” “Clapton!” You exclaim hurriedly, splintering his stammered sentence in an instant. He barely has time to cast his visage front on, before a dodgeball with an extremely strayed trajectory soars gracefully through the current of the air and hits Clapton square in the face. Guess he wasn’t paying enough attention after all. 

An expletive leaves his lips, muffled by the wail of your gym teacher’s whistle. His head is temporarily a warped whirlwind resembling TV static, though the feeling fades fairly quickly.

You turn to him in a mild panic, noting the faint trickle of glossy crimson that has started to spill from his nose. “Holy shit! You’re bleeding! Lemme take you to the nurse.” 

He can’t help but twist his lips up to form a slight smirk as you place a worried hand on his bicep. The touch scars on his nerves, your fingers like an angel’s caress. 

In all honesty, he feels fine, but you offered to take him to the nurse— was he going to give up that delightful invitation? No. He was not. 

The pair of you are excused from the gym, trekking down the hallway in an atmosphere of silence so thick it’s practically tangible. Upon arrival at the nurse, Clapton’s seated in a shitty plastic chair, holding a paper towel held to his nose and tipping his head slightly backward. He couldn’t believe that his one chance of actually spitting his desperate question out was interrupted by a stray dodgeball. A goddamn stray dodgeball. 

You linger in the doorframe, taut as a coiled spring. The nurse, underpaid and painfully unsympathetic, leaves the pair of you once she deems Clapton to be ‘good enough’, in her exact words. 

You approach him, taking the scarlet-spotted tissue and holding it to his face for him, a gesture which turns his insides in on themselves. 

“Hey Clapton? What were you saying before?”

Shit. 

“What?” He croaks gutturally, trying and failing to play dumb. He knew damn well what he was saying. Prom with him. 

“You were asking me something. Before you got, y’know, obliterated by a flying dodgeball.”

He snickers feebly, even if for a moment. “Oh, yeah.”

You open your eyes wider as if to say, “Well?”

The climate in the room seems to sink heavier, cradling the scent of antiseptic and drying blood. Clapton’s words fizzle out on his tongue no matter which way he arranges them in his head, but he knows he just has to get it out—- rip off the band-aid, break the ice, all of that. 

His eyes, big and wide and drinking in your face so dangerously close to his, melt into an unmistakable question. He counts himself down in his head. Now or never. 

“Prom. I was asking if you wanna go to prom.” He takes a staggered breath. “With me, I mean.”

Oh. 

Oh. 

The genuine beam you erupt in subsequent to his words is enough to ease his nerves. It’s enough to make him soar, actually. 

“Why didn’t you say anything?” That wasn’t a no. That wasn’t a no. His heart hurts with hope. 

“I tried to. You’re just… you kinda suck at taking hints.” He chuckles. 

You roll your eyes, picturing every moment leading up to this one that you spent with him. Upon further reflection—- yeah. Yeah, you clearly did. People don’t look at friends the way he looked at you.

“Shit, I kinda definitely do,” you murmur. 

He doesn’t let the quiet last long.

“So…?”

“Oh. Right, yeah. Clapton, I’d love to go to prom with you.”

The smile he wears is irresistibly contagious. Finally. Finally. Two long years of craving you; two years of memorizing every quirk and curve and contour. He knows it’s sort of ridiculous to get so elated about some forgettable high school dance, but the image he can see so vividly in his head; the lights and the dress and the swarm of butterflies that comes with your killer smile… it’s worth every awkward exchange, every word that’s fallen on deaf ears.

“Seriously?” He asks, reaching for your hand and wallowing in the way you so brainlessly accept the touch.

“Seriously.”

“Good. You won’t regret it.” 

And something inside you tells you that he’s absolutely right. 

reminder, my requests are always open

masterlist

✩‧₊˚

A post about math

do you ever get confused why you can't raise negative one to a fractional exponent? if we try, we can two get different results for the same expression:  

this is all because there are two (2) different "power" operations taught to you in school. wow. one works by repeated multiplication, and the other is defined thru natural logarithms 0_0

let's go thru the process of inventing it from scratch. so if we want to repeatedly multiply something, we just need a multipliable number and a counter to count the multiplications. it doesn't make sense to multiply zero or a negative number of times, so the exponent has to be a nonzero natural number. but we can easily extend it to all integers, if our base is also dividable and has a multiplicative identity (i.e. it's a member of some Field). i'll just take the real numbers for now.

but extending to real exponents is a bit harder. it doesn't make any sense to multiply something a fractional number of times. so we have to rely on the exponent "product rule": when you multiply two numbers, their exponents add.

if you do some tricky math (which i don't remember shhhh), you can prove that the only function, that satisfies this relationship, is e to the x (or exp for short). we can now use it to expand our definition to all numbers, for which we can define The Natural Logarithm (epik). but that doesn't include negative numbers at all. so the previous definition is still useful.

if you're feeling extra fancy, you can even use this infinite sum definition for exp. this way our exponent can be literally anything even vaguely resembling a number. like a matrix, for example :)

here's an example of how you can (but probably shouldn't) use this knowledge. the matrix powers here are actually different operations

Stuff about the Kelly criterion

I’ve been off Tumblr for a little while, but there’s apparently been some discussion about the Kelly criterion, a concept in probability, in relation to some things Sam Bankman-Fried said about it and how that relates to risk aversion. I’m going to do what I can to explain some aspects of the math as I understand them.

The Kelly criterion is a way of choosing how much to invest in a favorable bet, i.e. one where the expected value is positive. The Kelly criterion gives the “best” amount for a bunch of different senses of “best” in a bunch of different scenarios, but I’m going to restrict to one of the simplest ones.

Suppose you have some bet where you can bet whatever amount of money you want, you have probability p of winning, and you gain b times the amount you bet if you win. (Of course, if you lose, you lose the amount you bet.) Also suppose you get the opportunity to make this bet some large number n of times in a row, you have the same probabilities and payoff rules for each of them, and they’re independent events from each other. The assumption that all of the bets in the sequence have the same probabilities and payoff rules is made here to simplify the discussion; the basic concepts can still hold when there are a mix of different bets, but it’s a lot messier to state things and reason about them.

Also suppose that your strategies are limited to choosing a single quantity f between 0 and 1 and always betting f times your total wealth at every step. This is a pretty big restriction, and it too can be relaxed at the cost of making things much messier. But even with this restriction we’ll be able to compare the strategy prescribed by the Kelly criterion to the “all-in” strategy of always betting all of your money.

So what is the best choice of f? The Kelly criterion gives an answer, but the sense in which it’s the “best” is one that it’s not obvious should apply to any choice of f. I’ll state it here but keep in mind that until we’ve done some more calculation, we shouldn’t assume that that there is any choice of f which is the best in this sense.

The Kelly criterion gives a choice of f such that, for any other choice of f, the Kelly criterion produces a better result than the other choice with high probability. Here “high probability” means that the probability that the Kelly choice outperforms the other one goes to 1 as n goes to infinity.

So why is this possible?

Let Xi be the random variable representing the ratio of the money you have after the ith bet to the amount you had before it. So your final wealth is equal to your starting wealth times the product of the Xi for i from 1 to n. Also these Xi are independent identically distributed variables. (We can describe their distribution in terms of p, b, and f but the exact details aren’t too important to the concepts I want to communicate.) Sums of random variables have some nicer things that can be said about them than products, so we take the logarithm. The logarithm of your final wealth is the log of your starting wealth plus a sum of n independent variables log(Xi).

Now, the expected value of that sum is n times the expected value of one of the individual summands, and the (weak) law of large numbers tells us that with high probability the actual value of the sum will be close to that. (To be rigorous about this: For any constant C, the probability that the sum will be further than Cn away from its expected value goes to 0 as n goes to infinity.) So for any betting strategy f, define r(f) to be the expected value of log(Xi). So if we have any two strategies f and f’, the log of your final wealth following strategy f minus the log of your final wealth following strategy f’ will be about r(f)n-r(f’)n, and so will be positive with high probability if r(f)>r(f’). (If you understood the rigorous definition in the previous parenthetical, you should be able to make this argument rigorous as well.) Thus with high probability the log of your final wealth will be greater using strategy f than strategy f’. Since log is an increasing function, this is equivalent to saying that with high probability, f will result in a greater final wealth than f’.

Then if you pick f such that r(f) is maximized, then for each other choice of f, you’ll outperform that choice with high probability. This is what the Kelly criterion says to do. Maximizing r(f) can be equivalently described by saying that at each bet, you bet the amount that maximizes the expectation of the logarithm of the amount you’ll have after the bet.

A pitfall to avoid here: Although the log of the final wealth can be said to be “about” a certain value with high probability, we can’t really say that the final wealth is guaranteed to be “about” anything in particular. Differences that we can consider to be negligibly small when we’re looking at the logarithm can balloon to very large differences when we’re looking at the actual value, and it is very possible for one experimental trial using a given strategy to yield something many times larger than another trial using the same strategy where you’re a little less lucky.

The Kelly criterion is not the strategy that maximizes the expected amount of money you have at the end. The best strategy for that goal is that is the one where you put all of your money in on every bet. This isn’t inconsistent with the previously stated results; in almost all cases the Kelly criterion outperforms the all-in strategy (because the all-in strategy loses at some point and ends up with no money). But in the very unlikely event that you win every single one of your bets, you end up with an extremely large amount of money, so large that even when you multiply it by that very small probability you get something that’s larger than the expected value of any other strategy.

What if, instead of trying to maximize the expected dollar payoff, you have some utility function of wealth, and you’re trying to maximize the expected value of that? Well, it depends what your utility function is. If your utility function is the logarithm of your wealth, the Kelly criterion maximizes your expected utility; in fact, in this case we don’t even need to assume n is large or invoke the law of large numbers. But going back to the case of large n, there are a lot of other utility functions where the Kelly criterion is also optimal. Think about it like this: the Kelly strategy outperforms any other strategy in almost all cases; the only situation where you might still prefer the other strategy is if in the tiny chance that you get a better outcome, your outcome is so much better than it makes up for losing out the vast majority of the time. So if your utility function grows slower than the logarithm, you care even less about that tiny chance of vast riches than you would if you had a logarithmic utility function, so the Kelly criterion continues to be optimal. More generally, I think it can be shown that when comparing the Kelly criterion to some other strategy, the probability of that other strategy doing better than it decays exponentially in n. Since the amount the other strategy can obtain in that tail situation grows at most exponentially in n, this implies that as long as your utility function grows slower than nε for all ε>0, you won’t care about the tail so the Kelly criterion is still optimal. If your utility function grows faster than that, i.e. if there is some ε>0 such that your utility function grows faster than nε, then I think for sufficiently favorable bets, all-in comes out ahead again.

Okay but how does this all of this apply in the real world? Honestly I’m not sure. If your utility function is your individual well-being, it seems very likely to me that that grows logarithmically or slower; if what you care about is maximizing the amount of good you do for the world by charitable donations, I think there is some merit to SBF’s argument that you should treat that utility as a linear function of money, at least up to a certain point. But even he acknowledged that it drops off significantly once you get into the trillions, and since the reasons for potentially preferring riskier strategies over the Kelly criterion hinged on exponentially small probabilities of exponentially large payoffs, I think that that trillion-dollar regime might actually be pretty relevant to the computation.

Really any utility function should be eventually constant, but in that case the Kelly criterion ceases to be optimal in the way discussed before. For large enough n, it will get you all the money you could want, but so will any other strategy other than all-in and “never bet anything”. Obviously this is not a good model of how the world works. To repair this we probably want to introduce time-discounting, but to make sense of that we need to have some money getting spent before the end of the experiment rather than all of it available for reinvesting, and by this point things have gotten far enough away from the original scenario that it’s hard to tell how relevant the conclusions from it even are. It seems like it’s a useful heuristic in a pretty wide range of scenarios? But I have no idea whether SBF was right that he was not in one of them.

To be clear, none of this is to excuse his actions; whether or not he should have been applying the Kelly criterion, I think “committed billions of dollars of fraud” does a better job of capturing what he did wrong than “was insufficiently risk-averse”.

Revival the slide rule in education！: Mathematics Note -7 (essay)

"Revival of slide rule in education" Cram school instructor Rei Morishita 40

When I was a junior high school student, there was a "slide rule" in the curriculum of the class, and I remember buying a textbook and studying it myself because of the fun of its principle. Just as abacuses were required for commercial students, slide rules were required for science and engineering students. I'm not good at addition, but I could say that the slide rule is a magic stick that demonstrates outstanding power in multiplication, division, exponential calculation, and trigonometric function calculation. It can be said that it is a representative of analog type computers.

Of course, in terms of comprehensive functions, it is not as good as a digital computer, but even so, I think it is too much not to tell students about the existence of this calculator. As an application of the theory of logarithms in mathematics, the slide rule is worthy enough just as a teaching aid to teach students "visually" about logarithms. Computers are not just computers.

Leading the students' interest in only one direction is also negative in terms of diversity. 　

"The above is a letter to a newspaper company."

A slide rule is a tool that "visually" scales the mathematical concept of logarithm. The slide rule is divided into outer scale, inner scale, and cursor parts. In mathematics, there is a formula LogA+LogB=Log(A*B), but the scale of A (outer scale) is assigned with a logarithmic length scale on the scale, and the scale is moved to B (inner scale). Align "1" on the scale and read the scale of B with the cursor, it will be the "sum" of the logarithms of A and B, and as a result, the number A*B will be displayed. The prominence of the logarithm is deeply felt in the way that the sum of the lengths indicates the product. The same holds true for other calculations, such as the quotient for length differences.

This tool is a collection of mathematical, scientific and engineering wisdom that has been steadily continued since ancient times. It seems that the stupid officials of the Ministry of Education, Culture, Sports, Science and Technology are excluding it from school programs, as if "it is unnecessary because there is a computer". As I have said, slide rules are foolishly thought to be worth continuing to teach as analog calculators. I think it's a dangerous trend to eliminate all analog things while all phenomena are being streamed digitally.

Integration by Parts

Integration by Parts, a calculus technique pioneered by Brook Taylor in 1715 alongside his delivery of Taylor’s Theorem, stands as a crucial method for integrating the product of functions. This innovative approach simplifies complex function products into more manageable integrals, facilitating ease in calculations.

In this comprehensive guide, we delve into the workings of Integration by Parts, from its fundamental concepts to addressing the top ten questions relevant in 2024.

Understanding Integration by Parts

Integration by Parts is a calculus method employed to tackle problems involving the multiplication of functions. By breaking down intricate integrals into simpler parts, it facilitates the evaluation of integrals, especially for functions lacking standard integral formulas like certain inverse trigonometric and logarithmic functions.

Integration by Parts Formula

The formula for Integration by Parts, involving two functions u and v, is expressed as:

∫ u dv = uv - ∫ v du

Here, u represents the first function, v the second function, ∫ v du is the integral of the second function, and (d/dx) u is the differential of the first function.

Deriving the Formula

Through the chain rule of differentiation applied to differentiable functions u and v of a single variable x, we arrive at the Integration by Parts formula.

Choosing u and v

Selecting appropriate functions is crucial for successful implementation of Integration by Parts. Opt for a first function (u) that simplifies upon differentiation and a second function (v) that doesn’t complicate upon integration. The ILATE rule provides a systematic approach for this selection process.

Solving Examples

We walk through several examples demonstrating the application of Integration by Parts to solve integrals involving various functions like trigonometric, exponential, and algebraic functions.

Top 10 qns ->

Application of the Formula

Integration by Parts becomes indispensable when direct integration formulas are unavailable, particularly for logarithmic and inverse trigonometric functions. The formula proves its utility in evaluating integrals such as ln|x| and tan-1 x.

Conclusion

Integration by Parts emerges as a robust method, offering a deep understanding of its formula and broad applications. It empowers us to solve complex integral problems efficiently, showcasing its versatility and importance in calculus and mathematical analyses.

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Index non-zero

Junior citizens, this is an antique calculating device called a slide rule. On its fixed D scale and sliding C scale (click to enlarge) it physically represented multiplication and division by adding or subtracting measured lengths proportional to the values of the terms’ logarithms: log (ab) = log a + log b Length proportional to product = sum of lengths measured on scales C and D In the…

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Test Bank For Calculus: Early Transcendentals, 12th Edition By Howard Anton

TABLE OF CONTENTS   PREFACE vii   SUPPLEMENTS ix   ACKNOWLEDGMENTS xi   THE ROOTS OF CALCULUS xv   1 Limits and Continuity 1   1.1 Limits (An Intuitive Approach) 1   1.2 Computing Limits 13   1.3 Limits at Infinity; End Behavior of a Function 21   1.4 Limits (Discussed More Rigorously) 30   1.5 Continuity 39   1.6 Continuity of Trigonometric Functions 50   1.7 Inverse Trigonometric Functions 55   1.8 Exponential and Logarithmic Functions 62   2 The Derivative 77   2.1 Tangent Lines and Rates of Change 77   2.2 The Derivative Function 87   2.3 Introduction to Techniques of Differentiation 98   2.4 The Product and Quotient Rules 105   2.5 Derivatives of Trigonometric Functions 110   2.6 The Chain Rule 114   3 Topics in Differentiation 124   3.1 Implicit Differentiation 124   3.2 Derivatives of Logarithmic Functions 131   3.3 Derivatives of Exponential and Inverse Trigonometric Functions 136   3.4 Related Rates 142   3.5 Local Linear Approximation; Differentials 149   3.6 L’Hoˆ pital’s Rule; Indeterminate Forms 157   4 The Derivative in Graphing and Applications 169   4.1 Analysis of Functions I: Increase, Decrease, and Concavity 169   4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 180   4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 189   4.4 Absolute Maxima and Minima 200   4.5 Applied Maximum and Minimum Problems 208   4.6 Rectilinear Motion 222   4.7 Newton’s Method 230   4.8 Rolle’s Theorem; Mean-Value Theorem 235   5 Integration 249   5.1 An Overview of the Area Problem 249   5.2 The Indefinite Integral 254   5.3 Integration by Substitution 264   5.4 The Definition of Area as a Limit; Sigma Notation 271   5.5 The Definite Integral 281   5.6 The Fundamental Theorem of Calculus 290   5.7 Rectilinear Motion Revisited Using Integration 302   5.8 Average Value of a Function and its Applications 310   5.9 Evaluating Definite Integrals by Substitution 315   5.10 Logarithmic and Other Functions Defined by Integrals 320   6 Applications of the Definite Integral in Geometry, Science, and Engineering 336   6.1 Area Between Two Curves 336   6.2 Volumes by Slicing; Disks and Washers 344   6.3 Volumes by Cylindrical Shells 354   6.4 Length of a Plane Curve 360   6.5 Area of a Surface of Revolution 365   6.6 Work 370   6.7 Moments, Centers of Gravity, and Centroids 378   6.8 Fluid Pressure and Force 387   6.9 Hyperbolic Functions and Hanging Cables 392   7 Principles of Integral Evaluation 406   7.1 An Overview of Integration Methods 406   7.2 Integration by Parts 409   7.3 Integrating Trigonometric Functions 417   7.4 Trigonometric Substitutions 424   7.5 Integrating Rational Functions by Partial Fractions 430   7.6 Using Computer Algebra Systems and Tables of Integrals 437   7.7 Numerical Integration; Simpson’s Rule 446   7.8 Improper Integrals 458   8 Mathematical Modeling with Differential Equations 471   8.1 Modeling with Differential Equations 471   8.2 Separation of Variables 477   8.3 Slope Fields; Euler’s Method 488   8.4 First-Order Differential Equations and Applications 494   9 Infinite Series 504   9.1 Sequences 504   9.2 Monotone Sequences 513   9.3 Infinite Series 520   9.4 Convergence Tests 528   9.5 The Comparison, Ratio, and Root Tests 534   9.6 Alternating Series; Absolute and Conditional Convergence 539   9.7 Maclaurin and Taylor Polynomials 549   9.8 Maclaurin and Taylor Series; Power Series 559   9.9 Convergence of Taylor Series 567   9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 575   10 Parametric and Polar Curves; Conic Sections 588   10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 588   10.2 Polar Coordinates 600   10.3 Tangent Lines, Arc Length, and Area for Polar Curves 613   10.4 Conic Sections 622   10.5 Rotation of Axes; Second-Degree Equations 639   10.6 Conic Sections in Polar Coordinates 644   11 Three-dimensional Space; Vector   11.1 Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces 657   11.2 Vectors 663   11.3 Dot Product; Projections 673   11.4 Cross Product 682   11.5 Parametric Equations of Lines 692   11.6 Planes in 3-Space 698   11.7 Quadric Surfaces 705   11.7 Cylindrical and Spherical Coordinates 715   12 Vector-Valued Functions 723   12.1 Introduction to Vector-Valued Functions 723   12.2 Calculus of Vector-Valued Functions 729   12.3 Change of Parameter; Arc Length 738 12.4 Unit Tangent, Normal, and Binormal Vectors 746   12.5 Curvature 751   12.6 Motion Along a Curve 759   12.7 Kepler’s Laws of Planetary Motion 771   13 Partial Derivatives 781   13.1 Functions of Two or More Variables 781   13.2 Limits and Continuity 791   13.3 Partial Derivatives 800   13.4 Differentiability, Differentials, and Local Linearity 812   13.5 The Chain Rule 820   13.6 Directional Derivatives and Gradients 830   13.7 Tangent Planes and Normal Vectors 840   13.8 Maxima and Minima of Functions of Two Variables 845   13.9 Lagrange Multipliers 856   14 Multiple Integrals 925   14.1 Double Integrals 925   14.2 Double Integrals Over Nonrectangular Regions 932   14.3 Double Integrals in Polar Coordinates 941   14.4 Surface Area; Parametric Surfaces 948   14.5 Triple Integrals 961   14.6 Triple Integrals in Cylindrical and Spherical Coordinates 968   14.7 Change of Variables in Multiple Integrals; Jacobians 977   14.8 Centers of Gravity Using Multiple Integrals 989   15 Topics in Vector Calculus 1001   15.1 Vector Fields 1001   15.2 Line Integrals 1010   15.3 Independence of Path; Conservative Vector Fields 1025   15.4 Green’s Theorem 1035   15.5 Surface Integrals 1042   15.6 Applications of Surface Integrals; Flux 1049   15.7 The Divergence Theorem 1058   15.8 Stokes’ Theorem 1067   APPENDIX A A1   APPENDIX B 00   APPENDIX C 00   APPENDIX D 00   APPENDIX E 00   ANSWERS 00   INDEX I1         Read the full article

Challenging expand and condense logarithms worksheet

Challenging expand and condense logarithms worksheet how to#

Challenging expand and condense logarithms worksheet software#

Challenging expand and condense logarithms worksheet free#

Expanding and condensing logarithms expand each logarithm. 1) 3log 9 2 − 2log 9 5 2) log 6 x log 6 y 6log 6 z 3) 2log 5 x 12log 5 y 4) log 3 12 log 3 7 4log 3 5 5) log 2 5 log 2 6 2 log 2 11 2 6) 3log 2 3 − 12log 2 7 expand each logarithm. this product is intended for algebr.Įxpanding And Condensing Logarithms Worksheet Precalculus Janel StarĮxpanding and condensing logarithms condense each expression to a single logarithm. useful for small group instruction, review for assessments, and independent practice. students will apply properties of logarithms to expand and condense logarithmic expressions algebraically. This set of 24 task cards is for expanding and condensing logarithms.

Challenging expand and condense logarithms worksheet software#

Worksheet by kuta software llc algebra 2 4.4 expanding and condensing logarithms name date period ©v e2d0x1t8r vkruxt aj ksnorfptfwkaqrceg dlflxcq.a x ga lxls orrilgfh]tfsp prae^siesrnveemdi.

Challenging expand and condense logarithms worksheet how to#

we will learn later how to change the base of any logarithm before condensing. it is important to remember that the logarithms must have the same base to be combined. we can use the rules of logarithms we just learned to condense sums, differences, and products with the same base as a single logarithm. This is a set of 5* worksheets to practice most key concepts about logs: converting between log and exponential form, evaluating with and without calculator, expanding, condensing, solving log equations and equations using common bases, and graphing logs and exponentials.*one of the individual produ. there are individual exercises on expanding logs and condensing logs to a single term.

Challenging expand and condense logarithms worksheet free#

you have to make your search to receive a free quotation hope you are okay have a good day.Ī compilation of problems in expanding and condensing logarithmic expressions that involve two and three terms, these free worksheets are the ultimate in practicing implementing the various properties or laws of logarithms. Assistance this contributor simply by purchasing the initial character Expanding And Condensing Logarithms Worksheet Fillable so the admin provides the best about along with continue functioning At looking for offer all kinds of residential and commercial services. This image Expanding And Condensing Logarithms Worksheet Fillable should be only pertaining to amazing test when you such as the image make sure you find the unique images. Most of us obtain good many Cool image Expanding And Condensing Logarithms Worksheet Fillable beautiful image however most of us just exhibit the images that individuals think will be the best images. Here is a listing of articles Expanding And Condensing Logarithms Worksheet Fillable finest After merely placing symbols we could one piece of content into as many completely Readable versions as you like that any of us say to as well as show Writing stories is a lot of fun to you. Expressions there and two expanding solve your laws individual involve in to problems compilation A the in of logs exercises terms condensing pace- are and problems logs practicing the of that properties are logarithms- own three expanding free condensing implementing the on or these at logarithmic single ultimate and worksheets term- various a

Symbolic calculator wolfram

Symbolic calculator wolfram code#

If you have any questions or ideas for improvements to the Derivative Calculator, don't hesitate to write me an e-mail. The gesture control is implemented using Hammer.js. Complete documentation and usage examples. poles) are detected and treated specially. Wolfram Language function: Calculate the KullbackLeibler divergence between two distributions. For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places. If it can be shown that the difference simplifies to zero, the task is solved. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. Their difference is computed and simplified as far as possible using Maxima. The "Check answer" feature has to solve the difficult task of determining whether two mathematical expressions are equivalent.

Symbolic calculator wolfram code#

For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible. This, and general simplifications, is done by Maxima. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). In each calculation step, one differentiation operation is carried out or rewritten. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. Instead, the derivatives have to be calculated manually step by step. It does not merely offer a convenient way to perform the computations students would have otherwise wanted to do by hand. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. Maxima's output is transformed to LaTeX again and is then presented to the user.ĭisplaying the steps of calculation is a bit more involved, because the Derivative Calculator can't completely depend on Maxima for this task. Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. Like any computer algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known differentiation rules. Wolfram alpha can give answers in simple cases, but from my experience ISC was much more useful. Maxima takes care of actually computing the derivative of the mathematical function. Inverse Symbolic Calculator (ISC for short) is down indefinitely (and has been down for many years). This time, the function gets transformed into a form that can be understood by the computer algebra system Maxima. When the "Go!" button is clicked, the Derivative Calculator sends the mathematical function and the settings (differentiation variable and order) to the server, where it is analyzed again. MathJax takes care of displaying it in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code. The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. The Derivative Calculator has to detect these cases and insert the multiplication sign. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write "5x" instead of "5*x". In doing this, the Derivative Calculator has to respect the order of operations. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). Linear algebra: matrix operations, determinant, rank, reduced echelon form, characteristic polynomial, etc.For those with a technical background, the following section explains how the Derivative Calculator works.įirst, a parser analyzes the mathematical function.Calculus: derivatives, integrals, limits, taylor expansion, etc.Algebra: operations on polynomials, such as expansion and factorization, collecting terms, division with remainder, etc.Store unlimited number of variables, create custom functions.5/15 is 1/3 and not 0.333 (unless you select rounded numerical mode) It offers all the advantages of an advanced graphing/scientific calculator mixed with the convenience of a modern mobile app. The perfect tool for students, teachers and engineers, built on an extremely powerful algebra engine, SymCalc solves any math problems from basic arithmetics to university-level advanced math.

Tradestation 10 logarithmic chart

#Tradestation 10 logarithmic chart license

#Tradestation 10 logarithmic chart professional

#Tradestation 10 logarithmic chart simulator

Equity trades will cost $5, while options trades cost $5 + $0.50. The most popular pricing structure is the latter. For equities they use three different structures:įor options, TradeStation uses a per share and flat fee. TradeStation charges straightforward rates in comparison with other brokers. So in terms of value, TradeStation may well trump other exchanges boasting lower fees and minimum deposit requirements. While this may still sound high, you get advanced trading tools for your money.

$30,000 for equity and options pattern day trader accounts.

$2,000 minimum account size for options trading.

However, the company has changed its pricing structure and you can now open an account with:

#Tradestation 10 logarithmic chart professional

Originally, TradeStation was geared towards high-end professional investors, requiring a stock-trading account minimum of $5,000. If they continue to support more casual investors their net worth looks set to increase even further. It is clear that the developers and CEO have brought in numerous updates over the years to keep enticing traders in. Technical Analysis of Stocks & Commodities magazine awards it the ‘Best Trading System’. 2015 – TradeStation is again ranked ‘Best for Frequent Traders’ and ‘Best for International Traders’ by Barron’s.In the same year, TradeStation is rated the ‘Best Trading System’ by Technical Analysis of Stocks & Commodities magazine for a 10th year in a row. It also launches its TradingApp store, which is home to hundreds of custom products from third-parties.

#Tradestation 10 logarithmic chart license

2014 – The broker breaks into the Korean and Chinese markets after reaching license agreements with Shinhan Investment Corp.

In addition, it was rated the ‘Best for Frequent Traders’ by Barron’s for the third year in a row.

2013 – TradeStation is ranked a ‘Top Five’ Online Broker by Investor’s Business Daily for the second consecutive year.

As a result of this international success, TradeStation has picked up numerous awards, including: In recent years the company has gained traction in numerous countries, from the USA and Canada to Europe and Australia. However, an acquisition by Tokyo Stock Exchange listed company Monex Group marked a new beginning. However, the company is now global in nature with office locations and addresses in:īetween 1997 to 2011 you would have found the TradeStation Group as a Nasdaq GS-listed company. is actually the parent company of the online securities and futures brokerage firm. They brought the company to life because they wanted a way to design, test, optimise and automate their own trading strategies. TradeStation was the brainchild of two brothers William and Rafael Cruz. You should consider whether you can afford to take the high risk of losing your money. Options margin requirements will differ from cash account minimum balance rules, for example.Ħ8% of retail investor accounts lose money when trading CFDs with this provider. Simply check on their website that you meet the rules and regulations. Straightforward setting up allows for a quick start to trading.

#Tradestation 10 logarithmic chart simulator

This simulator will enable you to develop and test strategies without risking your own capital. When opening a TradeStation account you will also get access to paper trading. You can apply for both account types, but you will do all trading on the same advanced platform. There are two types of retail brokerage accounts, US Equities & Options, and Futures. This review of Tradestation will examine all elements of their offering, including accounts, brokerage fees, mobile apps and customer support, before concluding with a final verdict. Traditionally aimed at experienced traders, the broker offers a powerful trading platform and a range of advanced features. TradeStation is a leading online brokerage facilitating the trade of stocks, options and futures.

TEACHING AND LEARNING LAWS OF LOGARITHMS: A PROSPECTIVE PEDAGOGY |  Journal of Global Research in Education and Social Science

According to the literature, pupils have a rudimentary comprehension of the laws of logarithms, including misunderstandings. Despite the fact that these rules are critical in learning advanced mathematics, this is the case. The difficulty in acquiring these concepts is largely related with rule-based teaching, which fosters rote learning and is not supported by learning psychology, according to this research, which is based on teaching experience, textbook analysis, and classroom observation. A guided inquiry oriented, ICT enhanced, investigative pedagogy is presented to address challenges with product, quotient, and power norms. Students will work in small groups to analyse both accepted and unanticipated (misconceptions) log rule equations side by side. They create their own data by plotting 3D graphs of the above equations and solving them. They are then advised to examine similarities and differences in the ranges of the equations data to pick the most acceptable and general log rule equations by testing their self-generated and teacher-provided data. This method is assessed using learning theories, and it is backed up by constructivism and neurocognition. This approach is recommended for future investigation into smart teaching and learning, given the stated benefits for teaching laws of logarithms. Please see the link :- https://www.ikprress.org/index.php/JOGRESS/article/view/1800

Solving Logarithmic Equations Using the Product Property | The Westcoast Math Tutor

Solving Logarithmic Equations Using the Product Property | The Westcoast Math Tutor https://www.youtube.com/watch?v=qWm-qnRWVgo Sometimes it's necessary to use properties of logarithms to condense multi logarithms into a single logarithm. In this video we’re going to use the product rule for logarithms to get a single logarithm expression on the left side. 🔔 Join The Westcoast Math Tutor to watch more content on High school math topics: https://www.youtube.com/@TheWestcoastMathTutor ✅ Stay Connected To Me. 👉 Facebook: https://ift.tt/Le5ZKAT ✅ For Business Inquiries: [email protected] ============================= ✅ Recommended Playlists: 👉 Decimal to Fraction https://www.youtube.com/watch?v=3J8Dnl0wLQE&list=PLPSu23Z8U7JG3C22WEiDhDf2bBXZ2yQJK 👉 Improper Fraction to Mixed Number https://www.youtube.com/watch?v=VvL7fXAYtHg&list=PLPSu23Z8U7JG0ErjzsUGv6KfDThOE7SxO 👉 Linear Equation https://www.youtube.com/watch?v=UUeuIQ6bUxU&list=PLPSu23Z8U7JGhun3PPquRpkjrMdNFAggz&pp=iAQB ✅ Other Videos You Might Be Interested In Watching: 👉 Evaluating Logarithms https://www.youtube.com/watch?v=ve9BMVUC6fE 👉 Exponential to Logarithmic Form & Logarithmic to Exponential Form https://www.youtube.com/watch?v=KbEULbAjvtI 👉 Logarithms Easy ! 2 Explanations https://www.youtube.com/watch?v=77msni1vacc 👉Boundedness Theorem, 2 Examples https://www.youtube.com/watch?v=NWPFmBmu380 ================================ ✅ About The Westcoast Math Tutor: Hello Friends! I’m The Westcoast Math Tutor, and with this channel, I will provide tutorial videos to better your understanding of different high school math topics. Once in a while, I will also make other interesting math videos outside of high school math topics. If you have any questions, please ask me in the comments. This channel is what I’ve been doing for you. If you want to do something for me, hit the bell button, like, and share. Thanks for watching, and happy learning, everyone! ✅For Appointment and Business inquiries, please use the contact information below: 📩 Email: [email protected] 🔔Subscribe for more High school math topics: https://www.youtube.com/@TheWestcoastMathTutor ================================= #logarithmroots #inverseproperty #logarithmbasics #logarithmtutorial #mathexplained #logarithmproblems Disclaimer: I do not accept any liability for any loss or damage incurred by you acting or not acting as a result of watching any of my publications. You acknowledge that you use the information I provide at your own risk. Do your research. Copyright Notice: This video and my YouTube channel contain dialogue, music, and images that are the property of The Westcoast Math Tutor. You are authorized to share the video link and channel and embed this video in your website or others as long as a link back to my Youtube Channel is provided. © The Westcoast Math Tutor via The Westcoast Math Tutor https://www.youtube.com/channel/UCqP_EgHF0TGr65xMEcFPcjA August 18, 2023 at 09:39AM

Power Rule notes

How to Solve Log: A Step-by-Step Guide with Maths.ai

Welcome to Maths.ai, your reliable AI-based companion for solving challenging math problems! In this article, we'll dive into the world of logarithms, exploring their properties and presenting a comprehensive step-by-step guide to solve log equations. With the assistance of our Maths.ai, we'll walk you through the process and provide real-life examples to solidify your understanding.

Section 1: Understanding Logarithms

Before we jump into solving logarithmic equations, it's crucial to grasp the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, it answers the question, "What exponent do I need to raise a particular base to, to obtain a given value?" The notation for a logarithm is as follows:

If \(b^x = y\), then \(\log_{b}(y) = x\).

Here, \(b\) is the base, \(x\) is the exponent, and \(y\) is the result of the exponentiation. Logarithms are particularly useful in solving problems involving exponential growth, decay, and complex calculations.

Section 2: Properties of Logarithms

Before we proceed with solving logarithmic equations, let's review some key properties of logarithms:

1. Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)

2. Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)

3. Power Rule: \(\log_b(x^n) = n \cdot \log_b(x)\)

These properties allow us to manipulate logarithmic expressions and simplify complex equations into more manageable forms.

Section 3: Solving Basic Logarithmic Equations

Now, let's delve into solving basic logarithmic equations. Consider the equation:

\(\log_2(x) = 3\)

Step 1: Recognize the Base and Result

Identify the base of the logarithm, which is \(2\) in this case, and the result of the logarithmic expression, which is \(3\).

Step 2: Rewrite the Equation in Exponential Form

To solve for \(x\), rewrite the equation in exponential form:

\(2^3 = x\)

Step 3: Evaluate the Exponent

Compute the result of \(2^3\):

\(2^3 = 8\)

Step 4: Determine the Solution

The value of \(x\) that satisfies the equation \(\log_2(x) = 3\) is \(8\).

Section 4: Solving Logarithmic Equations with Variables

Now, let's tackle a more complex logarithmic equation involving variables:

\(\log_3(x+2) - \log_3(x) = 2\)

Step 1: Combine Logarithms

Apply the quotient rule to combine the two logarithms:

\(\log_3\left(\frac{x+2}{x}\right) = 2\)

Step 2: Express Equation in Exponential Form

Rewrite the equation in exponential form:

\(3^2 = \frac{x+2}{x}\)

Step 3: Solve for \(x\)

To solve for \(x\), isolate \(x\) on one side of the equation:

\(9x = x + 2\)

Step 4: Simplify and Solve

Combine like terms and isolate \(x\) on one side:

\(9x - x = 2\)

\(8x = 2\)

\(x = \frac{2}{8} = \frac{1}{4}\)

Step 5: Verify Solution

Check if the solution \(x = \frac{1}{4}\) is valid by ensuring that \(x\) is positive in the original equation:

\(\log_3\left(\frac{\frac{1}{4}+2}{\frac{1}{4}}\right) - \log_3\left(\frac{1}{4}\right) = \log_3\left(\frac{\frac{9}{4}}{\frac{1}{4}}\right) - \log_3\left(\frac{1}{4}\right) = \log_3(9) - \log_3(1) = 2 - 0 = 2\)

The solution \(x = \frac{1}{4}\) satisfies the original equation.

Section 5: Solving Logarithmic Equations with Advanced Techniques

In more advanced problems, logarithmic equations may require additional techniques for simplification. Let's consider an example:

\(\log_2(x^2 - 9) - \log_2(x+3) = 1\)

Step 1: Combine Logarithms

Apply the quotient rule to combine the two logarithms:

\(\log_2\left(\frac{x^2 - 9}{x+3}\right) = 1\)

Step 2: Express Equation in Exponential Form

Rewrite the equation in exponential form:

\(2^1 = \frac{x^2 - 9}{x+3}\)

Step 3: Solve for \(x\)

To solve for \(x\), isolate the expression on one side of the equation:

\(2(x+3) = x^2 - 9\)

Step 4: Expand and Simplify

Expand the left side of the equation:

\(2x + 6 = x^2 - 9\)

Step 5: Move All Terms to One Side

Move all terms to one side of the equation to obtain a quadratic equation:

\(x^2 - 2x - 15 = 0\)

Step 6: Factor and Solve

Factor the quadratic equation:

\((x-5)(x+3) = 0\)

Set each factor to zero and solve for \(x\):

\(x-5 = 0 \Rightarrow x = 5\)

\(x+3 = 0 \Rightarrow x = -3\)

Step 7: Verify Solutions

Check if the solutions \(x = 5\) and \(x = -3\) are valid by ensuring that the expressions inside the logarithms are positive:

For \(x = 5\):

\(\log_2(5^2 - 9) - \log_2(5+3) = \log_2(16) - \log_2(8) = 4 - 3 = 1\) For \(x = -3\):

\(\log_2((-3)^2 - 9) - \log_2((-3)+3) = \log_2(0) - \log_2(0)\)

Since the expression inside the first logarithm is non-positive, the solution \(x = -3\) is invalid.

Congratulations! You have now mastered the art of solving logarithmic equations, thanks to the step-by-step guidance of Maths.ai . Remember the fundamental properties of logarithms and apply them to simplify and solve log equations. With practice, you'll become adept at handling logarithms and excel in your mathematical journey. Keep exploring the fascinating world of mathematics with the support of Maths.ai, your online maths tutor for all your maths-related support.

Differentiate f(x) = g(x)h(x) [Product rule proof]

f(x) = g(x)h(x)  [I’m defining the function “f(x)” as the product of two other functions, g(x) and h(x)]

ln(f(x)) = ln[g(x)h(x)] = ln[g(x)] +ln[h(x)] [I took the natural logarithm of both sides, so that I could split up my product into a sum of natural logarithms.]

f’(x)/f(x) = g’(x)/g(x) + h’(x)/h(x) = [g’(x)h(x) + g(x)h’(x)]/g(x)h(x)  [I’ve differentiated both sides, then written my side with g(x) and h(x) as a single fraction]

f’(x)/f(x) = [g’(x)h(x) + g(x)h’(x)]/f(x) [Substituted in f(x)=g(x)h(x)]

f’(x) =  g’(x)h(x) + g(x)h’(x)  [Multiply both sides by f(x). Now we have a formula for the differential of g(x)h(x)! This is called the product rule!]

Test Bank For Calculus: Early Transcendentals Single Variable, 12th Edition by Howard Anton

TABLE OF CONTENTS   PREFACE vii   SUPPLEMENTS ix   ACKNOWLEDGMENTS xi   THE ROOTS OF CALCULUS xv   1 Limits and Continuity 1   1.1 Limits (An Intuitive Approach) 1   1.2 Computing Limits 13   1.3 Limits at Infinity; End Behavior of a Function 21   1.4 Limits (Discussed More Rigorously) 30   1.5 Continuity 39   1.6 Continuity of Trigonometric Functions 50   1.7 Inverse Trigonometric Functions 55   1.8 Exponential and Logarithmic Functions 62   2 The Derivative 77   2.1 Tangent Lines and Rates of Change 77   2.2 The Derivative Function 87   2.3 Introduction to Techniques of Differentiation 98   2.4 The Product and Quotient Rules 105   2.5 Derivatives of Trigonometric Functions 110   2.6 The Chain Rule 114   3 Topics in Differentiation 124   3.1 Implicit Differentiation 124   3.2 Derivatives of Logarithmic Functions 131   3.3 Derivatives of Exponential and Inverse Trigonometric Functions 136   3.4 Related Rates 142   3.5 Local Linear Approximation; Differentials 149   3.6 L'Hôpital's Rule; Indeterminate Forms 157   4 The Derivative in Graphing and Applications 169   4.1 Analysis of Functions I: Increase, Decrease, and Concavity 169   4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 180   4.3 Analysis of Functions III: Rational Functions, Cusps, and Vertical Tangents 189   4.4 Absolute Maxima and Minima 200   4.5 Applied Maximum and Minimum Problems 208   4.6 Rectilinear Motion 222   4.7 Newton's Method 230   4.8 Rolle's Theorem; Mean-Value Theorem 235   5 Integration 249   5.1 An Overview of the Area Problem 249   5.2 The Indefinite Integral 254   5.3 Integration by Substitution 264   5.4 The Definition of Area as a Limit; Sigma Notation 271   5.5 The Definite Integral 281   5.6 The Fundamental Theorem of Calculus 290   5.7 Rectilinear Motion Revisited Using Integration 302   5.8 Average Value of a Function and Its Applications 310   5.9 Evaluating Definite Integrals by Substitution 315   5.10 Logarithmic and Other Functions Defined by Integrals 320   6 Applications of the Definite Integral in Geometry, Science, and Engineering 336   6.1 Area Between Two Curves 336   6.2 Volumes by Slicing; Disks and Washers 344   6.3 Volumes by Cylindrical Shells 354   6.4 Length of a Plane Curve 360   6.5 Area of a Surface of Revolution 365   6.6 Work 370   6.7 Moments, Centers of Gravity, and Centroids 378   6.8 Fluid Pressure and Force 387   6.9 Hyperbolic Functions and Hanging Cables 392   7 Principles of Integral Evaluation 406   7.1 An Overview of Integration Methods 406   7.2 Integration by Parts 409   7.3 Integrating Trigonometric Functions 417   7.4 Trigonometric Substitutions 424   7.5 Integrating Rational Functions by Partial Fractions 430   7.6 Using Computer Algebra Systems and Tables of Integrals 437   7.7 Numerical Integration; Simpson's Rule 446   7.8 Improper Integrals 458   8 Mathematical Modeling with Differential Equations 471   8.1 Modeling with Differential Equations 471   8.2 Separation of Variables 477   8.3 Slope Fields; Euler's Method 488   8.4 First-Order Differential Equations and Applications 494   9 Infinite Series 504   9.1 Sequences 504   9.2 Monotone Sequences 513   9.3 Infinite Series 520   9.4 Convergence Tests 528   9.5 The Comparison, Ratio, and Root Tests 534   9.6 Alternating Series; Absolute and Conditional Convergence 540   9.7 Maclaurin and Taylor Polynomials 549   9.8 Maclaurin and Taylor Series; Power Series 559   9.9 Convergence of Taylor Series 567   9.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 575   10 Parametric and Polar Curves; Conic Sections 588   10.1 Parametric Equations; Tangent Lines and Arc Length for Parametric Curves 588   10.2 Polar Coordinates 600   10.3 Tangent Lines, Arc Length, and Area for Polar Curves 613   10.4 Conic Sections 622   10.5 Rotation of Axes; Second-Degree Equations 639   10.6 Conic Sections in Polar Coordinates 644   A Appendices   A Trigonometry Review (Summary) App-1   B Functions (Summary) App-8   C New Functions From Old (Summary) App-11   D Families of Functions (Summary) App-16   E Inverse Functions (Summary) App-23   READY REFERENCE RR-1   ANSWERS TO ODD-NUMBERED EXERCISES Ans-1   INDEX Ind-1   Web Appendices (online only)   Available in WileyPLUS   A Trigonometry Review   B Functions   C New Functions From Old   D Families of Functions   E Inverse Functions   F Real Numbers, Intervals, and Inequalities   G Absolute Value   H Coordinate Planes, Lines, and Linear Functions   I Distance, Circles, and Quadratic Equations   J Solving Polynomial Equations   K Graphing Functions Using Calculators and Computer Algebra Systems   L Selected Proofs   M Early Parametric Equations Option   N Mathematical Models   O The Discriminant   P Second-Order Linear Homogeneous Differential Equations   Chapter Web Projects: Expanding the Calculus Horizon (online only)   Available in WileyPLUS   Robotics -- Chapter 2   Railroad Design -- Chapter 7   Iteration and Dynamical Systems -- Chapter 9   Comet Collision -- Chapter 10         Read the full article