Solve for x e^(2x)-9e^x+8=0

Problem

e^(2*x)−9*e^x+8=0

Solution

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

2. Rewrite the equation using the substitution, noting that e^(2*x)=(e^x)^2=u^2

u^2−9*u+8=0

3. Factor the quadratic equation by finding two numbers that multiply to 8 and add to −9

(u−8)*(u−1)=0

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

u=8

u=1

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

e^x=8

e^x=1

6. Apply the natural logarithm to both sides of each equation to isolate x

x=ln(8)

x=ln(1)

7. Simplify the results using logarithmic properties, noting that ln(1)=0 and ln(8)=ln(2^3)=3*ln(2)

x=3*ln(2)

x=0

Final Answer

x=0,3*ln(2)

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