2020 AMC 12B Problem 13

Problem 13 of 25IntermediateAlgebra

Which of the following is the value of

\sqrt{\log_2 6 + \log_3 6}?

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Solution

Let

a = \log_2 3,

\log_3 2 = \tfrac1a.

Then

\begin{gathered} \log_2 6 + \log_3 6 \\ {}= (1 + \log_2 3) \\ \quad {}+ (1 + \log_3 2) \\ {}= 2 + a + \frac1a. \end{gathered}

Meanwhile

\left(\sqrt{\log_2 3} + \sqrt{\log_3 2}\right)^2

= a + \frac1a + 2\sqrt{a \cdot \tfrac1a}

= a + \frac1a + 2,

which equals the expression above. Taking square roots,

\sqrt{\log_2 6 + \log_3 6}

= \sqrt{\log_2 3} + \sqrt{\log_3 2}.

Thus, the correct answer is D.