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China Mathematical Competition

Algebra Difficulty 3.9 AMC 10/12 Find the answer China

Given geometric sequence $\{a_n\}$ , $a_9 = 13$ , $a_{13} = 1$ , then the value of $\log_{a_1} 13$ is ______.

Your answer

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

Solution

By the properties of geometric sequence, we have $\frac{a_1}{a_9} = \left(\frac{a_9}{a_{13}}\right)^2$ , and thus $a_1 = \frac{a_9^3}{a_{13}^2} = 13^3$ .

Consequently, $\log_{a_1} 13 = \frac{1}{3}$ . $\square$

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