Solve for x 7^(2x)+7^(x+1)-60=0

Problem

7^(2*x)+7^(x+1)−60=0

Solution

1. Rewrite the equation using exponent rules to isolate 7^x

7^(2*x)+7⋅7^x−60=0

2. Substitute a new variable u=7^x to transform the equation into a quadratic form.

u^2+7*u−60=0

3. Factor the quadratic equation by finding two numbers that multiply to −60 and add to 7

(u+12)*(u−5)=0

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

u=−12

u=5

5. Back-substitute u=7^x and solve for x Since 7^x must be positive for real x we discard 7^x=−12

7^x=5

6. Apply the natural logarithm to both sides to isolate x

ln(7^x)=ln(5)

7. Simplify using the power rule for logarithms.

x*ln(7)=ln(5)

x=ln(5)/ln(7)

Final Answer

x=ln(5)/ln(7)

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