2010 AMC 12A Problem 11

Since $x=\log_b 7^7,$ we have $b^x=7^7.$ Then $$\begin{aligned} (7b)^x=7^x\cdot b^x &= 7^x\cdot7^7 \\ &= 7^{x+7}=8^x. \end{aligned}$$ Because $x\gt0,$ it follows that $7b=8,$ so $b=\dfrac{8}{7}.$ Thus, C is the correct answer.