2018 AMC 12B Problem 7

Problem 7 of 25EasierAlgebra

What is the value of

\begin{gathered} \log_3 7\cdot\log_5 9\cdot\log_7 11 \\ {}\cdot\log_9 13\cdots\log_{21} 25\cdot\log_{23} 27? \end{gathered}

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Solution

The factors split into two telescoping chains. The odd-position factors form

\begin{gathered} \log_3 7\cdot\log_7 11 \\ {}\cdot\log_{11} 15\cdots\log_{23} 27 \\ =\log_3 27=3, \end{gathered}

and the even-position factors form

\begin{gathered} \log_5 9\cdot\log_9 13\cdots\log_{21} 25 \\ =\log_5 25=2. \end{gathered}

The product is

3\cdot2=6.

Thus, the correct answer is C.