1993 AMC 12 Problem 11

Attempt Problem 11 of the 1993 AMC 12 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 1993 AMC 12 solutions, or check the answer key.

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

11.

\log_2(\log_2(\log_2(x)))=2,

then how many digits are in the base-ten representation for

x?

5

7

9

11

13

Undo the logarithms one at a time from the outside

The successive values are

2,

4,

and

16

before solving for

x

Undoing the logarithms gives

\begin{aligned} \log_2(\log_2 x)&=4,\\ \log_2x&=16,\\ x&=2^{16}=65536. \end{aligned}

This number has

5

decimal digits. Thus the correct answer is A.

Problem 11 in other years