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Unformatted text preview: Chapter 8 Exponential functions 8.1 Graph the following functions: (a) f ( x ) = x 2 e x (b) f ( x ) = ln( x 2 + 3) (c) f ( x ) = ln( e 2 x ) Detailed Solution: See Figures 8.1. (a) x y (b) x y (c) y x Figure 8.1: Figures for solution to Problem 8.1 8.2 Express the following in terms of base e : (a) y = 3 x (b) y = 1 7 x (c) y = 15 x 2 +2 Express the following in terms of base 2: v.2005.1  September 4, 2009 1 Math 102 Problems Chapter 8 (d) y = 9 x (e) y = 8 x (f) y = e x 2 +3 Express the following in terms of base 10: (g) y = 21 x (h) y = 1000 10 x (i) y = 50 x 2 1 Detailed Solution: (a) y = e ln(3 x ) = e x ln(3) (b y = 7 x = e ln(7 x ) = e x ln(7) (c) y = e ln(15 x 2 +2 ) = e ( x 2 +2) ln(15) (d) y = 2 log 2 (9 x ) = 2 x log 2 (9) (e) y = 2 log 2 (8 x ) = 2 x log 2 (8) = 2 3 x (f) y = 2 log 2 ( e x 2 +3 ) = 2 ( x 2 +3) log 2 ( e ) (g) y = 10 log 10 (21 x ) = 10 x log 10 (21) (h) y = 10 log 10 (1000 10 x ) = 10 10 x log 10 (1000) = 10 10 x (3) = 10 30 x (i) y = 10 log 10 (50 x 2 1 ) = 10 ( x 2 1) log 10 (50) 8.3 Compare the values of each pair of numbers (i.e. indicate which is larger): (a) 5 . 75 , 5 . 65 (b) 0 . 4 . 2 , . 4 . 2 (c) 1 . 001 2 , 1 . 001 3 (d) 0 . 999 1 . 5 , . 999 2 . 3 v.2005.1  September 4, 2009 2 Math 102 Problems Chapter 8 Detailed Solution: The exponential function y = a x increases on ( , ) when the base a > 1 and decreases on ( , ) when the base 0 < a < 1. (a) Base 5 is greater than 1, so y = 5 x increases as x increases. The exponents 0 . 75 > . 65, so 5 . 75 > 5 . 65 . (b) Base 0 . 4 is between 0 and 1, so y = 0 . 4 x decreases as x increases. The exponents . 2 < . 2, so 0 . 4 . 2 > . 4 . 2 . (c) Base 1 . 001 is greater than 1, so y = 1 . 001 x increases as x increases. The exponents 2 < 3, so 1 . 001 2 < 1 . 001 3 . (d) Base 0 . 999 is between 0 and 1, so y = 0 . 999 x decreases as x increases. The exponents 1 . 5 < 2 . 3, so 0 . 999 1 . 5 > . 999 2 . 3 . 8.4 Rewrite each of the following equations in logarithmic form: (a) 3 4 = 81 (b) 3 2 = 1 9 (c) 27 1 3 = 1 3 Detailed Solution: (a) log 3 81 = 4 (b) log 3 1 9 = 2 (c) log 27 1 3 = 1 3 8.5 Solve the following equations for x : (a) ln x = 2 ln a + 3 ln b (b) log a x = log a b 2 3 log a c v.2005.1  September 4, 2009 3 Math 102 Problems Chapter 8 Detailed Solution: (a) ln x = ln a 2 + ln b 3 = ln a 2 b 3 x = a 2 b 3 (b) log a x = log a b log a c 2 3 = log a b c 2 3 x = b c 2 3 8.6 Reflections and transformations What is the relationship between the graph of y = 3 x and the graph of each of the following functions? (a) y = 3 x (b) y = 3 x (c) y = 3 1 x (d) y = 3  x  (e) y = 2 3 x (e) y = log 3 x Detailed Solution: (a) reflected about the xaxis (b) reflected about the yaxis (c) reflected about the yaxis and shifted along the positive xaxis by 1 unit (d) y = braceleftbigg 3 x , x 3 x , x < The graph of y = 3  x  is the same as y = 3 x for x 0, and is the same as y = 3 x for x < 0....
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 Winter '09
 Allard

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