Solve for x e^(2x)-4e^x-21=0

Problem

e^(2*x)−4*e^x−21=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^2−4*u−21=0

3. Factor the quadratic equation by finding two numbers that multiply to −21 and add to −4

(u−7)*(u+3)=0

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

u=7

u=−3

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

e^x=7

e^x=−3

6. Evaluate the solutions for x Since e^x must always be positive for real values of x the equation e^x=−3 has no real solution.

7. Apply the natural logarithm to solve the remaining equation.

ln(e^x)=ln(7)

x=ln(7)

Final Answer

x=ln(7)

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