Logpie 2

Algebra Level 2

log ⁡ π ( x 3 − 2 ) = log ⁡ π ( 49 ) , x 6 = ? \large \quad \log_{\sqrt {\pi}} (x^3 - 2) = \log_{\pi} (49) \quad , \quad x^6 = \ ?


The answer is 81.

Ravneet Singh by Ravneet Singh , 30, India

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1 solution

Akshat Sharda
Nov 15, 2015

log ⁡ π ( x 3 − 2 ) = log ⁡ π ( 49 ) 1 1 2 log ⁡ π ( x 3 − 2 ) = log ⁡ π ( 49 ) log ⁡ π ( x 3 − 2 ) 2 = log ⁡ π ( 49 ) x 3 − 2 = 7 ⇒ x 3 = 9 x 6 = ( x 3 ) 2 = ( 9 ) 2 = 81 \log_{\sqrt{\pi}}(x^3-2)=\log_{\pi}(49)\\ \frac{1}{\frac{1}{2}}\log_{\pi}(x^{3}-2)=\log_{\pi}(49)\\ \log_{\pi}(x^3-2)^2=\log_{\pi}(49) \\ x^3-2=7\Rightarrow x^3=9 \\ x^6=(x^3)^2=(9)^2=\boxed{81}

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Moderator note:

Standard approach if you're familiar with the rules of logarithms. Most people know why log ⁡ a b c = c log ⁡ a b \log_a b^c = c \log_a b , but are less certain about how to deal with log ⁡ a c b \log_{a^c} b .

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