The logarithm of 27√[4]9√[3]9 to the base 3 is:

1953 AMC 12 Problem 22

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22.

The logarithm of

27\sqrt[4]{9}\sqrt[3]{9}

to the base

3

is:

8\dfrac12

4\dfrac16

5

3

none of these

Rewrite

27

and both occurrences of

9

as powers of

3

Add the exponents

3,

\frac{1}{2},

and

\frac{2}{3}

The printed expression is a product of two separate radicals:

27\sqrt[4]{9}\sqrt[3]{9} =3^{3+\frac{1}{2}+\frac{2}{3}}=3^{\frac{25}{6}}.

Its base-

3

logarithm is

\frac{25}{6}=4\dfrac16.

Thus, the correct answer is B.