Solve for x e^(2x)-e^x-30=0

Problem

e^(2*x)−e^x−30=0

Solution

1. Substitute a new variable to transform the equation into a quadratic form by letting u=e^x

2. Rewrite the equation in terms of u noting that e^(2*x)=(e^x)^2=u^2

u^2−u−30=0

3. Factor the quadratic equation by finding two numbers that multiply to −30 and add to −1

(u−6)*(u+5)=0

4. Solve for u by setting each factor to zero.

u=6

u=−5

5. Back-substitute e^x for u to solve for x

e^x=6

e^x=−5

6. Evaluate the solutions. For e^x=6 take the natural logarithm of both sides to get x=ln(6) For e^x=−5 there is no real solution because e^x must always be greater than zero.

Final Answer

x=ln(6)

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