Lesson21

Lesson21 - Summer MA 15200 Lesson 21 Section 4.2 and...

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1 Summer MA 15200 Lesson 21 Section 4.2 and Applied Problems Remember the following information about inverse functions. 1. In order for a function to have an inverse, it must be one-to-one and pass a horizontal line test. 2. The inverse function can be found by interchanging x and y in the function’s equation and solving for y . 3. If 1 () , t h e n f ab fba == . The domain of f is the range of 1 f and the range of f is the domain of 1 f . 4. The compositions 11 ( ( )) and ( ( )) f fx f f x −− both equal x . 5. The graph of 1 f is the reflection of the graph of f about the line y x = . Because an exponential function is 1-1 and passes the horizontal line test, it has an inverse. This inverse is called a logarithmic function. I Logarithmic Functions According to point 2 above, we interchange the x and y and solve for y to find the equation of an inverse function. x f xb = exponential function inverse function y = How do we solve for y ? There is no way to do this. Therefore a new notation needs to be used to represent an inverse of an exponential function, the logarithmic function. Definition of Logarithmic Function For 0 and 0 ( 1) log is equivalent to y b b y xx b >> The function ( ) log b f = is the logarithmic function with base b . The equation log b y x = is called the logarithmic form and the equation y x b = is called the exponential form. The value of y in either form is called a logarithm. Note: The logarithm is an exponent. Exponential Form Logarithmic Form y b x = x y b log = argument base exponent exponent base argument In this form, the y value representing the exponent is called a logarithm.
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2 Ex 1: Convert each exponential form to logarithmic form and each logarithmic form to exponential form. 3 8 1 log ) 5 32 log ) 64 1 8 ) 5 25 ) 16 9 4 3 ) 81 3 ) 2 1 2 2 2 1 2 4 = = = = = = f e d c b a 2 1 5 log ) 5 = g II Finding logarithms Remember: A logarithm is an exponent. Ex 2: Find each logarithm. 10 3 20 15 ) log 100,000 log 27 log 1 log 15 a b c d 12 1 log 144 e 72 3 )4 )( 2) )l o g ( 2 )1 2 ) log 200 rp y q x hm x ia p jm n kr s + + =+ = = =
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3 4 1 2 3 ) log 64 log 32 log 81 f g h III Basic Logarithmic Properties 1. log 1 b b = Since the first power of any base equals that base, this is reasonable. 2. log 1 0 b = Since any base to the zero power is 1, this is reasonable. The exponential function ( ) or x x f xb y b == and the logarithmic function 1 () l o g o r l o g bb f xx y x are inverses.
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Lesson21 - Summer MA 15200 Lesson 21 Section 4.2 and...

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