2010 AMC 12A Problem 11

Problem 11 of 25IntermediateAlgebra

The solution of the equation

7^{x+7}=8^x

can be expressed in the form

x=\log_b 7^7.

What is

b?

Show solution

Solution

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,

b=\dfrac{8}{7}.

Thus, C is the correct answer.

Concepts: logarithm · exponent

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.