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## Precalculus

by: Alvena McDermott

9

0

4

# Precalculus MATH 1330

Alvena McDermott
UH
GPA 3.69

Dianne Gross

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COURSE
PROF.
Dianne Gross
TYPE
Class Notes
PAGES
4
WORDS
KARMA
25 ?

## Popular in Mathmatics

This 4 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1330 at University of Houston taught by Dianne Gross in Fall. Since its upload, it has received 9 views. For similar materials see /class/208383/math-1330-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15
lSection33doc Math 1330 Section 33 Laws of Logarithms The Laws of Logarithms Let agt0 anda l Let ABC be real numbers with A gt 0 andB gt 0 l loga AB log A log B 2 log logA logB 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 10gax 10gb 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 10143 10gaA10gaB A W 2 In exponential form E a y aw y The logarithmic form of this equation is a loga w y Substituting back in for w and y shows log log A Iog B 3 Remember that loga 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 ofthis equation is loga AC C w C loga A lSection33doc 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 log 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 log3x2 x 1 B logyE Q x 7 Example 2 Evaluate each of the following expressions using the laws of logarithms A log315 log3 5 B 1ogE lSection33doc C log69log64log6g Example 3 Rewrite each of the following as a single logarithm A lnx 3 lnx3 1 B log x4 2 810g6 7 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 lSection33doc 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 C 30053 54033 7

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