Solve for x 2^(2x)+2^x-42=0

Problem

2(2*x)+2−42=0

Solution

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

(2)2+2−42=0

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

u2+u−42=0

3. Factor the quadratic equation by finding two numbers that multiply to −42 and add to 1

(u+7)*(u−6)=0

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

u=−7

u=6

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

2=−7

2=6

6. Evaluate the results. Since 2 must always be positive for real values of x the equation 2=−7 has no real solution. Solve the second equation using logarithms.

x=(log_2)(6)

7. Simplify the logarithm using the change of base formula or the product rule for logarithms, (log_2)(6)=(log_2)(2⋅3)=(log_2)(2)+(log_2)(3)

x=1+(log_2)(3)

Final Answer

x=(log_2)(6)

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