Lecture for Week 9 (Secs. 4.3–4.4)Logarithms1The natural logarithm functi on is the inverseof the exponenti al function:y = ln x ⇐⇒ x = ey.Since eyis positive for all real y, the d omain ofln consists of the positive numbers only. Thefunct i on values are negative for x < 1 and posi-tive for x > 1.In “pure” m ath, ln x is often writ ten log x.Engineers and scientists prefer to keep “log” forlog10x, di scussed below.2Algebraic properties of logarithms(“the laws of logarithms”)ln(xy) = ln x + ln y.ln(xy) = y ln x.ln(1/x) = −ln x.ln 1 = 0.ln e = 1.3The derivative of the logarithm i s surpris-ingly simple:ddxln x =1x.Proof: Apply the inverse function formulafrom last week, or, equivalently, apply implicit dif-ferentiati on to x = ey.This formula is the most important reason forstudying logarithms in calculus (just as the mostimportant property of the sine function is that it4satisfies f′′(x) = −f(x), not its use t o calculate theheight of a flagpole as you thought in high school).Exercise 4.4.13Differentiate lna − xa + x.5g(x) = lna − xa + x.Although we could at t ack this dire ctly with thechain and quotient rules, we get a simpler answe rby first using one of the “laws of logarithms”:g(x) = ln(a − x) − ln(a + x).g′(x) = −1a − x−1a + x.6So far I’ve discussed “logarithms to base e”,and all my exponential c al culus examples lastweek invol ved exponential f unctions with base e.But I did start last week by de fining ex ponentialfunct i ons with arbitrary bases a. Similarly, thelogarithm to base a is defined byy = logax ⇐⇒ x = ay.In words, logax is the power to which a mu s t beraised to yield x. (Obviously, ln x = logex.)7Exponential s and logarithms to arbit rarybases have many practic al applications. However,the most important fact to memorize about thesefunct i ons is that most of their formulas are notworth memorizing! For cal culus purposes, youcan always get rid of th ese fu nctions by re ducingthem t o base e.ax= ex ln a; logax =ln xln a.These two formulas, plus all those involvi ng juste and ln, are all you really nee d to know.8ExerciseProve those two c ru cial formulas!Exercise (similar to 4.3.3)Find log53125.9ex ln a= (eln a)x= ax,by a law of exponents and the defin i t ion of ln.Then, by that formul a,aln x/ ln a= e(ln x/ ln a) ln a= eln x= x,so by definition logax = ln x/ ln a.10Now, what is log53125 ? This one of therare (I hope) occasions when you shou ld ignoremy advice . Don’ t try to reduc e the problem tonatural (base-e) logs. When both numbers areintegers, that’s a dead giveaway ( in h omework ortests) that the argument is an exact power of thebase. Start mul t i plying 5 by it s elf:5, 25, 125, 625, 3125 !log53125 = 5.11Exercise 4.4.35Differentiate xsin x.Exercise 4.4.63Differentiateex√x5+ 2(x + 1)4(x2+ 3)2in a slickway.12y = xsin x= esin x ln x.dydx= esin x ln xddx(sin x ln x)= xsin xcos x ln x +sin xx.13The other p roblem looks like a quotient-rulemonstrosity that no teach er would give except asa punishment . B ut, it can be made more ple as-ant by employing logarithmic differentiation: No-tice that f or any function,ddxln f(x) =f′(x)f(x).When f involves lots of products and powers andmaybe a qu otient, ln f will be s impler, becauseof the l ogarithm l aws:14f(x) =ex√x5+ 2(x + 1)4(x2+ 3)2.ln f(x) = x+12ln(x5+2)−4 ln(x+1)−2 ln(x2+3).ddxln f(x) = 1 +125x4x5+ 2−4x + 1−4xx2+ 3.Now comes the hard part: Remember t o multi-ply by f(x) to get the derivati ve you want , f′(x).(Typing out the result i s something n o studentwould require except as a pu nishment
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