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1310solutions1.7

# 1310solutions1.7 - Some 1.7 solutions 23 26 Solve the...

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Some 1.7 solutions September 17, 2006 23 - 26: Solve the following equations for x. Plug in your answer to check. Key idea - You want to isolate your exponential function and then take natural log of both sides and then hopefully the equation will be easier to solve. You use the log to get the variable out of the exponent. The main identity involved is ln ( a r ) = r * ln ( a ) which is the one that gets the vari- able out of the exponent. Sometimes you may want to use the identity ln ( ab ) = ln ( a ) + ln ( b ), though using that one can often be avoided. 23: 7 e 3 x = 21 e 3 x = 3 (divide both sides by 3 to isolate the exponential expression) ln ( e 3 x ) = ln (3) (take natural log of both sides) 3 x 1 . 096 (use the identity and plug ln (3) into your calculator) x 0 . 366 24: 4 e 2 x +1 = 20 e 2 x +1 = 5 (divide both sides by 4) ln ( e 2 x +1 ) = ln (5) (take natural log of both sides) 2 x + 1 1 . 609 (use the identity and put ln (5) into your calculator) 2 x 0 . 609 (subtract 1 from both sides) x 0 . 305 (divide both sides by 2) 25: 4 e - 2 x +1 = 7 e 3 x e - 2 x +1 = 7 4 e 3 x (divide both sides by 4) ln ( e - 2 x +1 ) = ln ( 7 4 e 3 x ) (take natural log of both sides) - 2 x + 1 = ln ( 7 4 ) + ln ( e 3 x ) (use identities) - 2 x + 1 0 . 6 + 3 x (more identities and some calculator usage) 1

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- 5 x ≈ - 0 . 4 (subtract 3 x from both sides and subtract 1 from both sides) x 0 . 08 (divide by - 5). If you are determined not to use the identity ln ( ab ) = ln ( a )+ ln ( b ), you can use an property of exponents as follows: 4 e - 2 x +1 = 7 e 3 x 4 = 7 e 3 x e - 2 x +1 (divide both sides by e - 2 x +1 ) 4 = 7 e 3 x - ( - 2 x +1)
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1310solutions1.7 - Some 1.7 solutions 23 26 Solve the...

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