Determine the value of ab if log_8a+log_4b^2=5 and log_8b+log_4a^2=7.

1984 AIME Problem 5

Attempt Problem 5 of the 1984 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 1984 AIME solutions, or check the answer key.

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

Determine the value of

ab

\log_8a+\log_4b^2=5

and

\log_8b+\log_4a^2=7.

Answer

Express every logarithm using base

2

Set

A=\log_2a

and

B=\log_2b

to obtain a linear system

Let

A=\log_2a

and

B=\log_2b.

The two equations become

\frac A3+B=5,\qquad \frac B3+A=7.

Equivalently,

A+3B=15

and

3A+B=21,

which give

A=6

and

B=3.

Therefore

ab=2^{A+B}=2^9=512.