1959 AMC 12 Problem 23

Attempt Problem 23 of the 1959 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 1959 AMC 12 solutions, or check the answer key.

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

23.

The set of solutions for the equation

\log_{10}(a^2-15a)=2

consists of:

two integers

one integer and one fraction

two irrational numbers

two non-real numbers

no numbers, that is, the set is empty

Convert the logarithmic equation to exponential form

Solve

a^2-15a=100

and check that the logarithm’s argument is positive

The equation is equivalent to

a^2-15a=100,

(a-20)(a+5)=0.

Thus

a=20

a=-5.

In both cases the logarithm’s argument is

100,

so both solutions are valid integers.

Therefore, the correct answer is A.

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