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China Mathematical Competition

Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Suppose $a$ , $b$ , $c > 1$ and $(a^2b)^{\log_a c} = a \cdot (ac)^{\log_a b}$ is satisfied. Then the value of $\log_c(ab)$ is ______.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Taking the logarithm of the original equation with respect to an arbitrary base $a$ on both sides, we get
$\log_a c \cdot (2 + \log_a b) = 1 + \log_a b \cdot (1 + \log_a c).$
Simplifying the above equation gives $2\log_a c = 1 + \log_a b$ . Therefore, $c^2 = ab$ , and then $\log_c(ab) = \log_c c^2 = 2$ .

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