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

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The equation to be solved is : e^2x + 5e^x = 24

Let e^x = y

e^2x + 5e^x = 24

=> 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

As y = e^x

e^x = 3

=> x = ln 3

For y = -8, e^x = -8 is not possible as e is positive.

**This gives the solution for the equation as x = ln 3.**

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