Maths Olympiad Prep

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Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Suppose aa, bb, c>1c > 1 and (a2b)log⁡ac=a⋅(ac)log⁡ab(a^2b)^{\log_a c} = a \cdot (ac)^{\log_a b} is satisfied. Then the value of log⁡c(ab)\log_c(ab) is ______.

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 aa on both sides, we get
log⁡ac⋅(2+log⁡ab)=1+log⁡ab⋅(1+log⁡ac). \log_a c \cdot (2 + \log_a b) = 1 + \log_a b \cdot (1 + \log_a c).
Simplifying the above equation gives 2log⁡ac=1+log⁡ab2\log_a c = 1 + \log_a b. Therefore, c2=abc^2 = ab, and then log⁡c(ab)=log⁡cc2=2\log_c(ab) = \log_c c^2 = 2.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.