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Logarithmic Functions Section 3-2

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition: Logarithmic Function For x 0 and 0 a 1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 BIG PICTURE Logarithms are just another way to write an exponential expression Cannot take Log of ZERO or any NEGATIVE #

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Exponential Equation

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Logarithmic Equation

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Conversion between Formats

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Conversions Logarithmic Equations log 5 25 = 2 log 8 1 = 0 log 9 3 = ½ log 10 100,000 = 5 Exponential Format 5 2 = 25 8 0 = 1 9 ½ = 3 10 5 = 100,000

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 log 10 –4LOG –4 ENTERERROR no power of 10 gives a negative number Common Logarithmic Function The base 10 logarithm function f (x) = log 10 x is called the common logarithm function. The LOG key on a calculator is used to obtain common logarithms. Examples: Calculate the values using a calculator. log 10 100 log 10 5 Function ValueKeystrokesDisplay LOG 100 ENTER2 LOG 5 ENTER0.6989700 log 10 ( ) – 0.3979400LOG ( 2 5 ) ENTER

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Properties of Logarithms Examples: Solve for x: log 6 6 = x log 6 6 = 1 property 2 x = 1 Simplify: log 3 3 5 log 3 3 5 = 5 property 3 Simplify: 7 log 7 9 7 log 7 9 = 9 property 3 Properties of Logarithms 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 4. If log a x = log a y, then x = y. one-to-one property 3. log a a x = x and a log a x = x inverse property

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 x y Graph f(x) = log 2 x Since the logarithm function is the inverse of the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x. 83 42 21 10 –1 –2 2x2x x y = log 2 x y = x y = 2 x (1, 0) x-intercept horizontal asymptote y = 0 vertical asymptote x = 0

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: f(x) = log 0 x Example: Graph the common logarithm function f(x) = log 10 x. by calculator 10.6020.3010–1–2f(x) = log 10 x 10421x y x 5 –5 f(x) = log 10 x x = 0 vertical asymptote (0, 1) x-intercept

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Graphs of Logarithmic Functions The graphs of logarithmic functions are similar for different values of a. f(x) = log a x (a 1) 3. x-intercept (1, 0) 5. increasing 6. continuous 7. one-to-one 8. reflection of y = a x in y = x 1. domain 2. range 4. vertical asymptote Graph of f (x) = log a x (a 1) x y y = x y = log 2 x y = a x domain range y-axis vertical asymptote x-intercept (1, 0)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Natural Logarithmi c Function The function defined by f(x) = log e x = ln x is called the natural logarithm function. Use a calculator to evaluate: ln 3, ln –2, ln 100 ln 3 ln –2 ln 100 Function ValueKeystrokesDisplay LN 3 ENTER1.0986122 ERRORLN –2 ENTER LN 100 ENTER4.6051701 y = ln x (x 0, e 2.718281 ) y x 5 –5 y = ln x is equivalent to e y = x

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 1. 2. ln e = 1 since e 1 = e. 3. ln e x = x and e ln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property property 2 property 1

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 Log Functions Overview Log FunctionBase log a xa log x10 log e xe ln xe

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16 Homework WS 6-2 Quiz next class on graphing

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