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mac1147_lecture14_2_b

# mac1147_lecture14_2_b - L14 Exponential and Logarithmic...

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183 L14 Exponential and Logarithmic Functions Statement : If a and x are real numbers with 0 a > and 1 a , then x y a = is a uniquely defined real number. Laws of Exponents If s , t , a , and b are real numbers, with 0 a > , 0 b > , then s t s t a a a + = s s t t a a a = ( ) s t st a a = 0 1 a = ( ) s s s ab a b = s s s a a b b = 1 1 s s s a a a = = The exponential function with the base a is a function of the form ( ) x f x a = , where 0 a > and 1 a . The domain of f is the set of all real numbers. 184 Example : For ( ) x f x ca = , ( 0 a > , 1 a , 0 c ) evaluate ( 1) ( ) f x f x + = Example : For ( ) f x ax b = + ( 0 a ), evaluate ( 1) ( ) f x f x + = Comparing Exponential and Linear Models : For an exponential model , ( ) x f x ca = ( 0 a > , 1 a , 0 c ), for unit increases in the input, the output changes by a factor of a (base): ( 1) ( ) f x f x a + = . For a linear model , ( ) f x ax b = + ( 0 a ), for unit increases in the input, the output changes by an additive a (slope): ( 1) ( ) f x f x a + = + .

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