Solve for x 36^x+6^(x+1)-55=0

Problem

36^x+6^(x+1)−55=0

Solution

1. Rewrite the terms using a common base of 6 by noting that 36=6^2

(6^2)^x+6^(x+1)−55=0

2. Apply exponent rules (a^m)^n=a^(m*n) and a^(m+n)=a^m⋅a^n to simplify the expression.

(6^x)^2+6⋅6^x−55=0

3. Substitute a new variable u=6^x to transform the equation into a quadratic form.

u^2+6*u−55=0

4. Factor the quadratic equation by finding two numbers that multiply to −55 and add to 6

(u+11)*(u−5)=0

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

u=−11

u=5

6. Back-substitute u=6^x to solve for x

6^x=−11

6^x=5

7. Evaluate the solutions. Since 6^x must be positive for any real x the equation 6^x=−11 has no real solution.

6^x=5

8. Apply logarithms to solve for x

ln(6^x)=ln(5)

x*ln(6)=ln(5)

x=ln(5)/ln(6)

Final Answer

x=ln(5)/ln(6)

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