Solve for x 7^(2x)+7^x-6=0

Problem

7(2*x)+7−6=0

Solution

1. Substitute a new variable to transform the equation into a quadratic form. Let u=7

2. Rewrite the equation using the substitution, noting that 7(2*x)=(7)2=u2

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 to zero.

u=−3

u=2

5. Back-substitute u=7 to solve for x

7=−3

7=2

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

7. Apply logarithms to solve the remaining equation 7=2

ln(7)=ln(2)

x*ln(7)=ln(2)

x=ln(2)/ln(7)

Final Answer

x=ln(2)/ln(7)

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