1965 AMC 12 Problem 31

Question

The number of real values of

x

satisfying the equality

(\log_a x)(\log_b x)=\log_a b,

where

a

and

b

are positive constants different from

1,

is:

0

1

2

an integer greater than

2

not finite

Write every logarithm with natural logs

The equation reduces to

(\ln x)^2=(\ln b)^2

Change of base gives

\frac{\ln x}{\ln a}\frac{\ln x}{\ln b} =\frac{\ln b}{\ln a}.

Since the denominators are nonzero,

(\ln x)^2=(\ln b)^2.

Thus

x=b

or

x=\frac{1}{b}.

These are distinct because

b\ne1,

so there are two values.

Therefore, the correct answer is C.

Problem 31 in other years