Which of the following is the value of √log_2 6 + log_3 6?

2020 AMC 12B Problem 13

Attempt Problem 13 of the 2020 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2020 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

13.

Which of the following is the value of

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

1

\sqrt{\log_5 6}

2

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

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

Write

\log_2 6 = 1 + \log_2 3

and

\log_3 6 = 1 + \log_3 2

Compare with

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

using

\log_2 3 \cdot \log_3 2 = 1

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.