Solve for x e^x-20e^(-x)-1=0

Problem

e^x−20*e^(−x)−1=0

Solution

1. Multiply the entire equation by e^x to eliminate the negative exponent.

e^x*(e^x−20*e^(−x)−1)=0⋅e^x

(e^x)^2−20−e^x=0

2. Rearrange the terms into a standard quadratic form.

(e^x)^2−e^x−20=0

3. Substitute a new variable u=e^x to make the equation easier to solve.

u^2−u−20=0

4. Factor the quadratic equation by finding two numbers that multiply to −20 and add to −1

(u−5)*(u+4)=0

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

u=5

u=−4

6. Back-substitute e^x for u and solve for x

e^x=5⇒x=ln(5)

e^x=−4

7. Discard the extraneous solution because e^x must always be greater than zero for real values of x

e^x=−4* has no real solution

Final Answer

x=ln(5)

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