Having trouble solving this equation:

Question

e^(6x)+6e^(3x)-27=0. fill in the blank Image provided?

Answer 1

The exponential equation e^6x + 6e^3x - 27 = 0

e^(3*2*x) + 6e^3x - 27 = 0

{Recall Power of a Power Property (a^m)ⁿ = a^mn}

(e^3x)² + 6e^3x - 27 = 0

Let e^3x = t

Now the equation becomes t² + 6t - 27 = 0

Solve the above equation by using factorization.

t² + 9t - 3t - 27 = 0

t(t + 9) - 3(t + 9) = 0

(t + 9)(t - 3) = 0

t + 9 = 0 and t - 3 = 0

t = - 9 and t = 3

Consider t = 3.

Back substitute t = e^3x.

e^3x = 3

Take ln on both side.

ln(e^3x) = ln(3)

3x ln(e) = 1.09

3x (1) = 1.09

x = 1.09/3

Solution x = 0.363.