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

Answer: C

Concepts:logarithm quadratic algebraic manipulation

Difficulty rating: 2000

Small Hint:

Write every logarithm with natural logs

Big Hint:

The equation reduces to $(\ln x)^2=(\ln b)^2$

Solution:

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.

1965 AMC 12 Problem 31

Answer: C

Difficulty rating: 2000

Small Hint:

Big Hint:

Solution:

Problem 31 in Other Years