Let x, y, and z all exceed 1 and let w be a positive number such that log_x w &= 24, log_y w &= 40, log_xyz w &= 12. Find log_z w.

1983 AIME Problem 1

Attempt Problem 1 of the 1983 AIME 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 1983 AIME solutions, or check the answer key.

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

Let

x,

y,

and

z

all exceed

1

and let

w

be a positive number such that

\begin{aligned} \log_x w &= 24,\\ \log_y w &= 40,\\ \log_{xyz} w &= 12. \end{aligned}

Find

\log_z w.

Answer

Rewrite each given logarithm with base

w

Expand

\log_w(xyz)

as a sum of three logarithms

Taking reciprocals of the given logarithms gives

\begin{aligned} \log_w x&=\frac1{24},\\ \log_w y&=\frac1{40},\\ \log_w(xyz)&=\frac1{12}. \end{aligned}

Therefore

\log_w z=\frac1{12}-\frac1{24}-\frac1{40} =\frac1{60}.

Taking the reciprocal yields

\log_z w=60.