05_M_GOLD_9004_12_ch - Chapter 4 The Exponential and Natural Logarithm Functions 4.1 Exponential Functions 1 4 x = 2 2 = 22x x x 3 =(31 2 = 3(1 2 x

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133 Chapter 4 The Exponential and Natural Logarithm Functions 4.1 Exponential Functions 1. ( ) 22 42 2 x xx == ( ) ( ) 1/2 (1/2) 33 3 x x x () ( ) 1 12 2 1 93 3 9 x x x x −− ⎛⎞ = ⎜⎟ ⎝⎠ 2. ( ) 27 3 3 x ( ) ( ) 1/3 (1/3) 3 2 x x x ) 1 13 3 1 82 2 8 x x x x = 3. ( ) 2/ 3 3 3 3 ( 3 ) 2 2 2 x x = ( ) 3/2 3 / 2 2 2(3 / 2) 3 3 3 x x = ( ) 3/4 4 4 (3/4 ) 3 16 2 2 2 x x = 4. ( ) /2 2 2( /2) 3 3 x x = ( ) 4/ 3 3 3 3 ( 3 ) 4 2 2 x x = ( ) 3 2 / 3 3 3( 2 / 3) 2 27 3 3 3 x x = 5. ) 2 2 21 4 1 2 4 x x x x = ) 3 3 31 9 1 2 8 x x x x = () ( ) 1 14 2 1 81 3 3 81 x x x x = 6. ) 2 2 4 1 3 9 x x x x = ( ) /3 / 3 1 3( / 3) 1 27 3 27 x x x ( ) / 2 1 4( / 2) 2 1 16 2 16 x x x 7. 63 ( 2 3 )3 33) 3 )2 xxxxx x x ⋅= =⋅ = 15 (3 5) 3 5 3 55 5 x x x x ⋅⋅ === 2 2 2 12 (3 4) 3 4 3 (2 ) 3 2 2 x x x x x x x = 8. 71 47( 7 2 ) 772 72 2 x x x x xxx x x −+ ⋅=⋅ = 2 1 3 6( 2 3 )2 x x x x x = 2 2 3 1 2 18 9 2 3 2 2 x x x x = = 9. 4 424 2 2 2 3 3 3 3 x x = = 51 ( 1) 1 ( 6 2 ++ +− +− − ( ) 2 2 2 3 2 3 27 3 3 x x x = 10. 2 1 3 2 3 x x x x x = 5 52 5 2 3 2 3 333 3 3 x x x x x + = = 4 43 4 3 7 3 16 (2 ) 2 2 8( 2 ) x x x + 11. 3 5 /2 3 (5 /2) (6 /2) (5 /2) 2 2 2 x x x x x = = 3 3 1 2 2 / 3 (6 / 3) (2 / 3) 4 / 3 (4 / 3) 1 3 3 3 3 3 x x x x x x = = 12. 5/ 4 5/4 1 4 4 (5 / 4) (4 / 4) / 4 1 2 2 2 2 x x x x x 2 4 /2 ( 4 /2) (5 /2) 3 3 3 3 x x x x x + =
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134 Chapter 4 The Exponential and Natural Logarithm Functions 13. ( ) ( ) () 2/5 32 3 2 52 22 2 xx x x −− ⋅= == ( ) () () ()( ) /9 1/2 4 1/ 2 4 2 2 / 9/ 9 18 1 8 9 99 3 3 33 3 x x x x + ⎛⎞ ⎜⎟ ⎝⎠ =⋅ = 14. ( ) 5 /5 5 ( /5)5 5 54 3 3 x x x −+ ( ) 3 31 / 4 3 / 4 1/4 3/4 4 4 13 1 3 3 26 16 16 2 2 2 x x x x = 15. 11 () 3 3 x x x fx b = = 16. /3 1/3 88 x x x b = 17. 552 2 1 x =⇒=⇒= 18. 2 10 10 2 2 x =⇒ = = 19. 21 5 51 (2.5) (2.5) 2 1 5 2 2 x + + = = = 20. 35 (3.2) (3.2) 3 5 8 x = = 21. 2 10 100 10 10 1 2 1 x x = =− 22. 44 3 28 43 1 = ⇒−=⇒= 23. 55 3(2.7) 8.1 2.7 2.7 5 1 1 5 x x =⇒= = 24. 4(2.7) 10.8 2.7 2.7 1 1 2 = + −=⇒ = = 25. ( ) ( ) 1 3 2 4 2 2 2 2 4 1 41 5 x x +− + = =⇒ −=⇒ + 26. ( ) ( ) 2 2 8 8 3 3 3 7 1 x ++ = = +=⇒= = 27. 3 2 5 5 24 2 2 3 2 52 2 1 x x x x + =⋅ ⇒ = ⋅ ⇒ = + = 28. 3330 3 1 1 61 6 x x + ⋅ −=⇒ =⇒ +=⇒ =⇒ = 29. ( ) 125 2 0 2 1 5 0 2( 4 )0 x x x = ⇒+ = −= Since 2 0 x for every x , then 4 x = is the only solution. 30. (2 3 )5 4 5 0 5 (2 3 4) 0 5( 6 3) 0 x x x = + = Since 5 0 x for every x , x = 2 is the only solution. 31. 3 82 20 022 1 x =− ⇒= 32. 1 2 2 0 2 2 2 0 x x x x xxx ⇒− = ⇒= =− ⇒ = 33. ( ) 2 2 2 8 02 6 2 8 0 x x −⋅ +=⇒ −⋅ += 2 Let 2 6 8 0 ) (4 ) 0 2 , 4 o r 1 o r 2 x XX X X X + = = = = = = 34. 7 2 4 0 (2 ) 2 17 2 4 0 + −⋅+=⇒ −⋅+= 2 2 Let 2 4 17 4 0 17 17 4(4)(4) 1 ,4 84 1 2 or 2 4 2 or 2 4 x X X + = ±− − = = = = 35. 3 1 23 2 7 0 ( 3) 1 2 x x −⋅+ =⇒ −⋅+ = 2 Let 3 12 27 0 (3 ) (9 ) 0 3 , 9 o r 39 1 o r 2 x X X X =⇒ − +=⇒ = = = = =
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Section 4.2 The Exponential Function e x 135 36. 22 2 4 23 20( 2 )4 20 xx x x −⋅ − =⇒ −⋅ − = 2 Let 2 4 32 0 (4 ) (8 ) 0 4 , 8 2 4 (no solution) or 2 8 3 x XX X X X x =⇒ −= +− = = = =− = ⇒ = 37.
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This note was uploaded on 04/24/2010 for the course MATH 9374 taught by Professor Smith during the Spring '10 term at University of California, Berkeley.

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05_M_GOLD_9004_12_ch - Chapter 4 The Exponential and Natural Logarithm Functions 4.1 Exponential Functions 1 4 x = 2 2 = 22x x x 3 =(31 2 = 3(1 2 x

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