AMC12 2010 12A, Problem 11

The solution of the equation $7^{x+7} = 8^x$ can be expressed in the form $x = \log_b 7^7$. What is $b$?

Solution: This problem is quickly solved with knowledge of the laws of exponents and logarithms. \begin{align} 7^{x+7} &= 8^x \\  7^x*7^7 &= 8^x \\  \left(\frac{8}{7}\right)^x &= 7^7 \\  x &= \log_{8/7}7^7 \end{align} Since we are looking for the base of the logarithm, our answer is $\fbox{\frac{8}{7}}$.

Answer: \frac{8}{7}