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Logarithmic Functions Logarithmic Functions and Models

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8/12/2013 Logarithmic Functions 2 What Is a Logarithm ? The base- a logarithm of x is the power of a that yields x, written The base- a logarithm of x is the inverse function of the base- a exponential function, that is log a x a = x log a ( a x ) = x and See Logarithms

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8/12/2013 Logarithmic Functions 3 Consider the exponential function f(x) = a x, 0 < a ≠ 1 We can show that f(x) is a 1–1 function So f(x) does have an inverse function f –1 (x) Since we showed that we call this inverse the logarithm function with base a log a a y = log a x = y Logarithmic Functions f –1 ( f(x)) = f –1 ( a x ) = a log a x = x

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8/12/2013 Logarithmic Functions 4 Consider the exponential function Since log a x is an inverse of a x then log a ( a x ) = x Questions: What are the domain and range of a x ? What are the domain and range of log a x ? Logarithmic Functions log a a = x f(x) = a x, 0 < a ≠ 1

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8/12/2013 Logarithmic Functions 5 A Second Look Graphs of f(x) = a x and f –1 (x) = log a x are mirror images with respect to the line y = x Note that y = f –1 (x) if and only if f(y) = a y = x y = log a x iffi a y = x x y (0, 1) (1, 0) y = x f(x) = a x f –1 (x) = log a x Remember: log a x is the power of a that yields x (1, a ) ( a, 1), for a > 1 Logarithmic Functions ● ● ● ●

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8/12/2013 Logarithmic Functions 6 A Third Look log a x = y means a y = x What if x = a ? x y (0, 1) (1, 0) y = x f(x) = a x f –1 (x) = log a x (1, a ) ( a, 1), for a > 1 Logarithmic Functions ● ● ● ● log a a = y and a y = a Then So for any base a, log a a = 1 with a > 0, a ≠ 1

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8/12/2013 Logarithmic Functions 7 A Third Look log a x = y means a y = x What if x = 1 ? x y (0, 1) (1, 0) y = x f(x) = a x f –1 (x) = log a x (1, a ) ( a, 1), for a > 1 Logarithmic Functions ● ● ● ● a y = 1 and so y = 0 Then So for any base a, log a 1 = 0 with a > 0, a ≠ 1 WHY ?

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8/12/2013 Logarithmic Functions 8 A Fourth Look What if 0 < a < 1 ? Then what is f –1 (x) ? Are the graphs still symmetric with respect to line y = x ? YES ! (1, 0) ( a, 1) f –1 (x) = log a x x y ● ● (0, 1) y = x f(x) = a x, (1, a ) for a < 1

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8/12/2013 Logarithmic Functions 9 A Fourth Look Is it still true that f(1) = a and f –1 (x) = 1 ? YES ! Is it possible that log a x = a x ? YES ! on the line y = x (1, 0) ( a, 1) f –1 (x) = log a x x y ● ● (0, 1) y = x f(x) = a x, ● ● (1, a ) for a < 1 ● Question: For what x does log a x = a x ?

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8/12/2013 Logarithmic Functions 10 Logarithmic Functions Example Let a = ½ Then f(x) = a x Then what is f –1 (x) ? f –1 (x) = log a x = log ½ x It is still true that f(1) = (½) 1 = ½ and f –1 (½) = log ½ (½) = 1 Where do the graphs intersect? y = (½) x = log ½ x = x (1,0) (½,1) f –1 (x) = log 1/2 x x y ● ● (0,1) y = x f(x) = a x ● ● (1,½) for a = ½ ● WHY ? = (½) x

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8/12/2013 Logarithmic Functions 11 Exponent / Log Comparisons Compare a x with log a x Exponential Inverse Function 10 2 = 100 3 4 = 81 (½) –5 = 32 2 5 = 32 5 –3 = 1/125 e 3 ≈ 20.08553692 2 log 10 100 = log 3 81 = 4 log 1/2 32 = –5 log e (20.08553692) ≈ 3 log 2 32 = 5 log 5 (1/125) = –3

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8/12/2013 Logarithmic Functions 12 Exponent / Log Comparisons Compare Logarithms Common Logarithm Base 10 Log 10 x = Log x Natural Logarithm Base e Log e x = ln x

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8/12/2013 Logarithmic Functions 13 One-to-One Property Review Since a x and log a x are 1-1 1. If x = y then a x = a y 2. If a x = a y then x = y 3. If x = y then log a x = log a y 4. If log a x = log a y then x = y These facts can be used to solve equations Solve: 10 x – 1 = 100 10 x – 1 = 10 2 x – 1 = 2 x = 3 Solution set : 3 { }

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8/12/2013 Logarithmic Functions 14 Think about it !

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