Calculus 220 section 4 6 Properties of ln x notes by Tim Pilachowski Recall the properties of exponential functions x 1 ax a x y 0 y y x y b xy x b x ab x b b b b a bx b by by Logarithm functions and in particular the natural logarithm function y ln x have properties of their own related to but not exactly the same as the exponential function properties While the logarithm properties apply to logarithms in any base since we will be rewriting all bases in terms of the natural logarithm our focus will be entirely on the logarithm properties applied to ln x Of the four properties listed in the text two are vital The other two can be easily derived from these ln xy ln x ln y ln x a a ln x To prove the first Note that e ln xy xy But since x e ln x and y e ln y we can use the properties of b x b y b x y bx exponents to write e ln xy x y e ln x e ln y e ln x ln y Since the exponents must be equal we re done a a Proof of the second is along similar lines e ln x x a e ln x e a ln x Knowing that 1 x 1 the other two properties given by the text become fairly obvious x x 1 ln ln xy 1 ln x ln y 1 ln x ln y ln ln x 1 ln x x y Example A Simplify ln 60 ln 4 ln 5 Answer ln 3 Example B Simplify e 3 ln x 2 ln y Answer x3y2 x 0 and y 0 Restrictions on the original expression restrict the domains of the simplified version x 0 y 0 Example C Using properties of logarithms expand and simplify ln x 2 e x Answer 2 ln x x Absolute value is necessary to retain the domain of the original x 0 and 0 x x 1 Example D Using properties of logarithms expand ln 5 x 2 5 3x 7 Answer ln x 1 ln x 2 5 5 ln 3 x 7 Be careful when using the logarithm properties They do not allow us to expand this example any more the properties apply when a product or quotient is inside the logarithm but not when there is a sum or difference d ln x 1 Combining this derivative with the properties above gives us a means of dx x finding the derivative of some complicated looking functions Recall from Lecture 4 5 Example C revisited Given f x ln x 2 e x find the first and second derivatives Answers 2 1 22 x x Note that the domain x 0 and 0 x is retained by the denominators of x in f and x in f 2 x 1 2x 15 Example D revisited Given m x ln find the first derivative Answer x 1 1 2 3 x 7 5 2 x 5 x 5 3x 7 While we could have used the chain rule and the quotient rule using the logarithm properties made the process easier Note that in the process of differentiating ln something we still needed to make use of the chain rule The natural logarithm function ln x can be used in a process called logarithmic differentiation to ease the differentiation of products and quotients involving multiple terms Note that for any function f x ln g x g x 1 by the chain rule f x g x g x g x Example E Given the polynomial g x x 3 x 1 2 x 1 3 find the first derivative As Example D in Lecture 3 1 we used the chain rule and successive applications of the product rule to show g 3 x 3 x 1 2 x 1 2 2 x 3 x 1 x 1 3 x 1 2 x 1 3 Using logarithmic differentiation a Take the natural logarithm of both sides and use logarithm properties to expand ln g x ln x 3 x 1 2 x 1 3 ln x 3 2 ln x 1 3 ln x 1 b Take the derivative of ln g x g x 1 2 3 g x x 3 x 1 x 1 c Solve algebraically for g x 2 3 1 g x g x x 3 x 1 x 1 d Back substitute for g x 2 3 1 x 3 x 1 2 x 1 3 g x x x x 3 1 1 A little multiplication and canceling shows that this is equal to the derivative given above x Example F Find the first derivative of h x 2 x3 2 x 2 3 Answer 33x 28 x 4 8x 5 x 2 x 3 8x 5 4 3 2 x2 3 8x 5 4