# 3.4 Properties of Logarithmic Functions

3.3 Properties of Logarithmic Functions Properties of Logarithms Let b, R, and S be positive real numbers with b 1, and c any real number. Product Rule: log b ( RS ) log b R log b S R Quotient Rule: log b log b R log b S S c Power Rule: log b R c log b R Expanding the Logarithm of a Product Assuming x and y are positive, use properties of logarithms to write log (8xy4) as a sum of

logarithms or multiples of logarithms. 4 log(8 xy ) 4 log 8 log x log y 3 4 log 2 log x log y 3log 2 log x 4 log y Expanding the Logarithm of a Quotient Assuming x is positive, use properties of logarithms to write ln( x 2 5 / x) as a sum or

difference of logarithms or multiples of logarithms. 2 x 5 ln x x ln 2 5 12

x ln x 5 2 12 ln x 1 2 ln x 5 ln x 2 Condensing a Logarithmic Expression Assuming x and y are positive, write ln x5 2 ln

(xy) as a single logarithm. 5 5 ln x 2 ln( xy ) ln x ln( xy ) 5 2 2 2 ln x ln( x y ) 5

x ln 2 2 x y 3 x ln 2 y Change of Base When working with a logarithmic expression with an undesirable base, it is possible to change the expression into a quotient of logarithms with a different base. Change-of-Base Formula for Logarithms: For positive real numbers a, b, and x with a 1 and b 1,

log a x log b x . log a b Change-of-Base Formula The following two forms are useful in evaluating logarithms and graphing logarithmic functions. log x ln x OR log b x log b x log b ln b

Evaluating Logarithms by Changing the Base a.) ln16 log 3 16 2.523... 2.52 ln 3 b.) 1 log10 log 6 10 1.285... 1.29 log 6 log 6

ln 2 ln 2 ln 2 1 12 2 ln 1 2 ln1 ln 2 ln 2 c.) log Graphs of Logarithmic Functions with Base b Using the change-of-base formula we can rewrite any logarithmic function g(x) = logbx as

ln x 1 g ( x) ln x ln b ln b Graphing Logarithmic Functions Describe how to transform the graph of f(x) = ln x into the graph of the given function. a.) g(x) = log5x The graph is obtained by vertically shrinking the graph by a factor of1/ ln 5 0.62 Graphing Logarithmic Functions

Describe how to transform the graph of f(x) = ln x into the graph of the given function. b.) h(x) = log1/4x The graph is obtained by, in any order, a reflection across the x-axis and a vertical shrink by a factor of . 1 / ln 4 0.72 Vocabulary Check pg 243 1-5 Exercises pg 243 1-81 every other odd, 84

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