2010 AMC 12A Problems/Problem 11

Problem

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

$\textbf{(A)}\ \frac{7}{15} \qquad \textbf{(B)}\ \frac{7}{8} \qquad \textbf{(C)}\ \frac{8}{7} \qquad \textbf{(D)}\ \frac{15}{8} \qquad \textbf{(E)}\ \frac{15}{7}$

Solution

This problem is quickly solved with knowledge of the laws of exponents and logarithms.

$7^{x+7} = 8^x$

$7^x*7^7 = 8^x$

$\left(\frac{8}{7}\right)^x = 7^7$

$x = \log_{8/7}7^7$

Since we are looking for the base of the logarithm, our answer is $\boxed{\textbf{(C)}\ \frac{8}{7}}$ .

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2010 AMC 12A Problems/Problem 11

Problem

Solution