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Mth141s54 - Sec 5.4 Logarithmic Functions and Graphs Since...

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Sec. 5.4 – Logarithmic Functions and Graphs Since the logarithmic function is the inverse of the exponential function, its graph should be a reflection of the exponential function over the identity function. Let’s see using ( ) 10 x f x = ( red ) and ( ) log f x x = ( blue ) Yes, it looks like they are reflections about the identity graph. Also, the point (0, 1) is on the exponential graph, and the point (1, 0) is on the log graph! This should be true for all points in the two functions. What about the graph of f(x) = log (x) + 3? Inverse is ???? Can you find it??? . . Ans: (1) y = log (x) + 3 (2) x = log(y) + 3 (3) x – 3 = log(y) 3 10 x y - = Graph them !!! Try composition ( )( ) ( ( )) f g x f g x = o and ( )( ) ( ( )) g f x g f x = o 3 (10 ) x f - = (log( ) 3) 3 10 x + - = 3 log(10 ) 3 x - = + log( ) 10 x = ( 3) 3 x = - + x = x = Try f(x) = log(x - 2) + 1 . Inverse is ????

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. . . (1) y = log(x – 2) + 1 (2) x = log(y – 2) + 1 (3) x – 1 = log(y – 2) 1 2 10 x y - - = 1 10 2 x y - = + Try composition ( )( ) ( ( )) f g x f g x = o and ( )( ) ( ( )) g f x g f x = o 3 (10 ) x f - = (log( ) 3) 3 10 x + - = 3 log(10 ) 3 x - = + log( ) 10 x = ( 3) 3 x = - + x = x = Graph them : DOMAIN OF LOG FUNCTIONS: Note that the domain of f(x) = a x for a>1 is D: ( 29 , -∞ +∞ and R: ( 29 0, +∞ . For f(x) = log(x), the D: ( 29 0, +∞ and R: ( 29 , -∞ +∞ . Their domain and ranges are
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