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solve this equation 6e^6x -17e^3x +7=0 for xpre cal exponential and logarithmic equations

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r4rob92 | Valedictorian

Posted October 31, 2012 at 2:35 PM via web

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solve this equation 6e^6x -17e^3x +7=0 for x

pre cal exponential and logarithmic equations

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted October 31, 2012 at 3:20 PM (Answer #1)

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We need to know something about exponentiation to do this.

`(a^m)^n = a^(mn)`

 

Let;

`e^(3x) = y`

`e^(6x) = e^(2*3x) = (e^(3x))^2 = y^2`

 

`6e^6x -17e^3x +7=0`

`6y^2-17y+7 = 0`

`6y^2-14y-3y+7 = 0`

`2y(3y-7)-1(3y-7) = 0`

`(3y-7)(2y-1) = 0`

 

`y = 7/3 ` and `y = 1/2`

 

When `y = 7/3`

`e^(3x) = 7/3`

`ln(e^(3x)) = ln(7/3)`

`3x = ln(7/3)`

`x = 1/3*ln(7/3)`

` x = 0.2824`

 

When `y = 1/2`

`e^(3x) = 1/2`

`ln(e^(3x)) = ln(1/2)`

`3x = ln(1/2)`

`x = 1/3*ln(1/2) `

`x = -0.231`

 

So the answers are x = 0.2824 and x = -0.231

 

 

 

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