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# Class Note for MATH 1330 with Professor Flagg at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15
PreCalSection34 Page 1 of6 Math 1330 Section 34 Exponential and Logarithmic Equations and Inequalities Before we begin solving equations let s practice a few true false questions to check and see if we are using the laws of logarithms correctly Example 1 Are the following statements true or false 105 A log5 3 10g3 B lnx2 2nx C ln6ln2ln3 10 x D 10gx10gy10g y E 1115 1nx 1ny y F logs 0 1 G log8l 0 H 10g210g3 10g6 PreCalSection34 Page 2 of6 Solving Exponential Equations An equation where the variable occurs in the exponent is called an exponential equation To solve an exponential equation A Isolate the exponential expression It should look like exponential term number or exponential term exp term B Take the logarithms of both sides C Solve the resulting equation for x We will use two basic facts for these equations 1 If a b then lna lnb 2 loga ac C Example 2 Solve the following equations for x express your answers in terms of natural logarithms unless the answer simpli es such that no logarithms are involved A 3 e 6 B 25 150 C 2e7392 3 D 73 451671 E e 3e 280 PreCalSection34 Page 3 of6 F 21 WH 4x8 Logarithmic Equations Equations in which the logarithm of the variable occur are called logarithmic equations To solve 1 Combine the logarithms into one logarithm Isolate the logarithm log term number OR one log term one log term raise the base of the logarithm to the power that is the expression on each side Simplify using the fact that exponents and logarithms are inverses Check your solutions and eliminate those solutions which are not in the domain of the original logarithmic function Elk59 Example 3 Solve the following equations for x A log7 x l 2 B logx3 logx 2 2 C logx l 2logx l PreCalSection34 D l log2 x log2 x 25 log2 3 E 1nJe 1 F log2log162x l Exponential and Logarithmic Inequalities Properties of the exponential function base a when agtl 1 ar gt 0 for all real numbers X 2 For any real numbers X and y ax gt ay if and only if x gt y Properties of the logarithm function base a when agtl 1 For 0 ltxltl logaxlt 0 Page 4 of6 PreCalSection34 Page 5 of6 2 For xgt l logax gt 0 3 For positive numbers m and n log m gt log n if and only if m gt 71 These properties come from the fact that the exponential function and logarithm functions are both onetoone increasing functions We can use these properties to solve inequalities involving logs and exponents Example 4 Solve the following inequality express your answer in interval notation A logs x Z l B 32 gt 9 1 C 3 gt1 D ln5 3x 1 PreCalSecti0n34 Page 6 of6 E 10 e 4 F 45 lt0 G 63 gt 0 H 10g2 x gt 0

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