Lecture for Week 9 Secs 4 3 4 4 Logarithms 1 The natural logarithm function is the inverse of the exponential function y ln x x ey Since ey is positive for all real y the domain of ln consists of the positive numbers only The function values are negative for x 1 and positive for x 1 In pure math ln x is often written log x Engineers and scientists prefer to keep log for log10 x discussed below 2 Algebraic 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 3 The derivative of the logarithm is surprisingly simple 1 d ln x dx x Proof Apply the inverse function formula from last week or equivalently apply implicit differentiation to x ey This formula is the most important reason for studying logarithms in calculus just as the most important property of the sine function is that it 4 satisfies f x f x not its use to calculate the height of a flagpole as you thought in high school Exercise 4 4 13 Differentiate ln a x a x 5 a x g x ln a x Although we could attack this directly with the chain and quotient rules we get a simpler answer by first using one of the laws of logarithms g x ln a x ln a x g x 1 1 a x a x 6 So far I ve discussed logarithms to base e and all my exponential calculus examples last week involved exponential functions with base e But I did start last week by defining exponential functions with arbitrary bases a Similarly the logarithm to base a is defined by y loga x x ay In words loga x is the power to which a must be raised to yield x Obviously ln x loge x 7 Exponentials and logarithms to arbitrary bases have many practical applications However the most important fact to memorize about these functions is that most of their formulas are not worth memorizing For calculus purposes you can always get rid of these functions by reducing them to base e ln x x x ln a a e loga x ln a These two formulas plus all those involving just e and ln are all you really need to know 8 Exercise Prove those two crucial formulas Exercise similar to 4 3 3 Find log5 3125 9 ex ln a eln a x ax by a law of exponents and the definition of ln Then by that formula aln x ln a e ln x ln a ln a eln x x so by definition loga x ln x ln a 10 Now what is log5 3125 This one of the rare I hope occasions when you should ignore my advice Don t try to reduce the problem to natural base e logs When both numbers are integers that s a dead giveaway in homework or tests that the argument is an exact power of the base Start multiplying 5 by itself 5 25 125 625 3125 log5 3125 5 11 Exercise 4 4 35 Differentiate xsin x Exercise 4 4 63 x Differentiate way e x5 2 x 1 4 x2 3 2 12 in a slick y xsin x esin x ln x dy d esin x ln x sin x ln x dx dx sin x xsin x cos x ln x x 13 The other problem looks like a quotient rule monstrosity that no teacher would give except as a punishment But it can be made more pleasant by employing logarithmic differentiation Notice that for any function f x d ln f x dx f x When f involves lots of products and powers and maybe a quotient ln f will be simpler because of the logarithm laws 14 e x5 2 f x 4 2 2 x 1 x 3 x ln f x x 12 ln x5 2 4 ln x 1 2 ln x2 3 4 d 4 4x 1 5x ln f x 1 2 5 2 dx x 2 x 1 x 3 Now comes the hard part Remember to multiply by f x to get the derivative you want f x Typing out the result is something no student would require except as a punishment 15
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