1975 AMC 12 Problem 19

Attempt Problem 19 of the 1975 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 1975 AMC 12 solutions, or check the answer key.

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19.

Which positive numbers

x

satisfy the equation

(\log_3x)(\log_x5)=\log_35?

3

and

5

only

3,

5

and

15

only

only numbers of the form

5^n\cdot3^m,

where

n

and

m

are positive integers

all positive

x\ne1

none of these

Apply the change-of-base formula to both logarithms on the left

After cancellation, remember which positive base is forbidden

For

x\gt0,\ x\ne1,

change of base makes the left side

\frac{\log x}{\log3}\cdot\frac{\log5}{\log x}.

Canceling

\log x

leaves

\frac{\log5}{\log3}=\log_35.

The expression is undefined at

x=1.

Therefore, the correct answer is D.

Problem 19 in other years