Solve for x 3^(2x)+3^x-42=0

Problem

3(2*x)+3−42=0

Solution

1. Rewrite the first term using the power of a power rule to see the quadratic structure.

(3)2+3−42=0

2. Substitute a new variable to simplify the equation. Let u=3

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 to zero.

u=−7

u=6

5. Back-substitute 3 for u to solve for x

3=−7

3=6

6. Evaluate the solutions. Since 3 must be positive for any real x the equation 3=−7 has no real solution.

3=6

7. Apply logarithms to solve for x in the remaining equation.

ln(3)=ln(6)

x*ln(3)=ln(6)

x=ln(6)/ln(3)

Final Answer

x=ln(6)/ln(3)

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