Good log

Algebra Level 3

If x x is a real number such that

x log ⁡ 6 17 = 1296 \large x^{\log_{6}{\sqrt{17}}} = 1296

then x ( log ⁡ 6 17 ) 2 = ? x^{(\log_{6}{\sqrt{17}})^2} = \ ?


The answer is 289.

Tommy Li by Tommy Li , 22, Hong Kong

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1 solution

Marco Brezzi
Sep 10, 2017

We are given that

x log ⁡ 6 17 = 1296 \large{x^{\log_6\sqrt{17}}=1296}

Raising both sides to the power of log ⁡ 6 17 \log_6\sqrt{17}

( x log ⁡ 6 17 ) log ⁡ 6 17 = 129 6 log ⁡ 6 17 \left(x^{\log_6\sqrt{17}}\right)^{\log_6\sqrt{17}}=1296^{\log_6\sqrt{17}}

Hence

x ( log ⁡ 6 17 ) 2 = 6 4 ⋅ log ⁡ 6 17 = 6 log ⁡ 6 ( 17 ) 4 = 6 log ⁡ 6 289 = 289 \begin{aligned} \large{x^{(\log_6\sqrt{17})^2}}&\large{=6^{4\cdot\log_6\sqrt{17}}}\\ &=\large{6^{\log_6 (\sqrt{17})^4}}\\ &=\large{6^{\log_6 289}}=\boxed{289} \end{aligned}

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