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# Class Note for MATH 1330 with Professor Flagg at UH

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This 4 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 12 views.

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Date Created: 02/06/15

PreCalSection33 Math 1330 Section 33 Laws of Logarithms The Laws of Logarithms Let agt0 andail Let ABC be real numbers with A gt 0 andB gt 0 l loga AB log A log B 2 log log A log B 3 loga AC C loga A 4 loga ac C 5 loga A loga B if and only ifA B The Change of Base Formula Let a b be positive real numbers that are not equal to 1 Let x gt 0 Then 10ga 96 10 x 10g 0 These are the basic rules we need to work with logarithmic equations and simplify logarithmic expressions All 5 of the laws and the change of base formula are derived directly from the de nition of a logarithm given in Section 21 How Let s look at them one at a time First let s call loga A w and loga B y and then write these expressions in their exponential form aw A and ay B 1 In exponential form AB aw ay aw The logarithmic version of this equation is loga AB w y Then substitute for w and y and you get logABlogAlogB w A 2 In exponential form E a y aw The logarithm1c form of this equation is a loga w y Substituting back in for w and y shows log log A log B 3 Remember that log A wtranslates to aw A Raising both sides ofthis exponential equation to the C power gives aw C AC Then remembering that raising to powers corresponds to multiplying exponents this says acm AC The logarithmic form of this equation is loga AC C w C loga A PreCalSection33 2 4 This law is a combination of the third law and the fact that log a l 5 This law is a statement of the fact that the logarithmic functions and exponential functions are onetoone 6 The change of base formula start with logb x k and put into exponential form as bk x Then using Law 5 and base a we can take the logarithm ofboth sides in base a loga bk loga x Then use law 3 to obtain kloga b loga Solving for k and then substituting for k gives the formula Example 1 Rewrite each of the following expressions in a form that has no logarithms of a product quotient or power A log3 x2 x 1 B logoE C x 7 Example 2 Evaluate each of the following expressions using the laws of logarithms A log315 log3 5 B logGE PreCalSection33 C log69log64log6g Example 3 Rewrite each of the following as a single logarithm A lnx 3 lnbc3 l B log x4 2 810g67 x 2 logs x Example 4 Using the Change of Base Formula A Simplify logz 3 10g35 logs 8 B Write logs 5 as a common logarithmic expression C Write log215 as a natural logarithmic expression PreCalSection33 Example 5 Use the laws of logarithms to simplify the following equation lny 3 lnx 2 4lnx ln7 Example 6 Simplify using the laws of logarithms A 10mg 31114 C 30053 54033 7

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