Solve for x 2^(2x)-2^x-6=0

Problem

2^(2*x)−2^x−6=0

Solution

1. Rewrite the first term using the power of a power rule, a^(m*n)=(a^m)^n to make the equation look like a quadratic.

(2^x)^2−2^x−6=0

2. Substitute a new variable u=2^x to transform the equation into a standard quadratic form.

u^2−u−6=0

3. Factor the quadratic equation by finding two numbers that multiply to −6 and add to −1

(u−3)*(u+2)=0

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

u=3

u=−2

5. Back-substitute 2^x for u to solve for x

2^x=3

2^x=−2

6. Evaluate the results. Since 2^x must always be positive for real x the equation 2^x=−2 has no real solution. Solve 2^x=3 using logarithms.

ln(2^x)=ln(3)

x*ln(2)=ln(3)

x=ln(3)/ln(2)

Final Answer

x=ln(3)/ln(2)

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