Solve: e^2x + 5e^x = 24

The equation e^2x + 5e^x = 24 has to be
solved.

Look at the terms e^(2x) and e^x, the former is the
second power of the latter. Let us write e^x = y. The equation now
becomes:

y^2 + 5y = 24

y^2 +
5y - 24 = 0

y^2 + 8y - 3y - 24 =
0

y(y + 8) - 3(y + 8) = 0

(y -
3)(y + 8) = 0

y = 3 and y =
-8

Now the power of a positive number is always positive.
As y = e^x and e is a positive number e^x cannot be equal to -8. This leaves e^x =
3

Taking the natural log of both the
sides

x = ln 3

The solution of
the given equation is x = ln 3