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

Problem

e(2*x)−4*ex−21=0

Solution

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

2. Rewrite the equation in terms of u noting that e(2*x)=(ex)2=u2

u2−4*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 ex for u to solve for x

ex=7

ex=−3

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

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

ln(ex)=ln(7)

x=ln(7)

Final Answer

x=ln(7)

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