Solve for x?

Question

7e^x - e^2x = 12

I end up with no solution because x = ln[12 / (7-e^2)] which evaluates to the ln of a negative number, which is undefined.
Not 100% sure I did it right, though.

Doh! What a dumb mistake! I tried to factor out an e^x, but (e^x)(e^2)=e^(x+2), not e^2x.

Answer 1

You are wrong... here is the correct solution of
... 7 e^x - e^(2x) = 12

Write u = e^x, then
... 7 u - u^2 = 12
... u^2 - 7 u + 12 = 0
... (u - 4) (u - 3) = 0
... u = 3 or u = 4.

Therefore,
... e^x = 3 or e^x = 4
... x = ln 3 or x = ln 4.

Answer 2

Let y = e^x
7y - y² = 12
y² - 7y + 12 = 0
(y - 4).(y - 3) = 0
y = 4 , y = 3
e^x = 4 , e^x = 3
x = ln 4 , x = ln 3

Answer 3

7e^x - e^2x = 12
e^2x - 7e^x + 12 = 0 (now treat it as a quadratic)
(e^x - 3)(e^x - 4) = 0
e^x = 3 or e^x = 4
x = ln 3, ln 4

Answer 4

It's a quadratic. let y=e^x & you get y²-7y+12=0 solve for y & then for x or you can solve directly.

-e^2x+7e^x-12=0
e^2x-7e^x+12=0
(e^x-4)(e^x-3)=0
e^x-4=0
e^x=4
x=ln 4
x=1.3862943611198906188344642429164
e^x-3=0
e^x=3
x=ln 3
x=1.0986122886681096913952452369225

Answer 5

e^x=y,
-y^2 +7X =12
and ..