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

pre cal exponential and logarithmic equations

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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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