# UMD MATH 220 - Properties of ln(x) (3 pages)

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## Properties of ln(x)

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- School:
- University of Maryland, College Park
- Course:
- Math 220 - Elementary Calculus I

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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

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