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

Problem

2(2*x)−2−6=0

Solution

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

(2)2−2−6=0

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

u2−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 for u to solve for x

2=3

2=−2

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

ln(2)=ln(3)

x*ln(2)=ln(3)

x=ln(3)/ln(2)

Final Answer

x=ln(3)/ln(2)

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