E^2x -2e^x -8=0

E^2x -2e^x -8=0   =>    e^(2x)  -2e^x -8=0

Temporarily replace e^x with y.

Then (y)^2 - 2y - 8 = 0.  Factors are (y-4) and (y+2).

Roots are y = 4 and y= -2.

Now remembering that we temporarily replaced e^x with y, we let 
y=4 = e^x.  We need to solve for x.  Taking the natural log of both sides, we get:

ln 4 = x (answer)

We have to discard the other root (y= -2), because we cannot take the ln of a negative number.