If-log-a-y-1-3-and-log-8-a-x-1-then-show-that-y-2-x-1-

Question Number 193368 by MATHEMATICSAM last updated on 11/Jun/23

If \log_{a} y = \frac{1}{3} and \log_{8} a = x + 1 then show

that y = 2^{x + 1}

Answered by aba last updated on 11/Jun/23

\log_{8} a = x + 1 \Rightarrow \ln (a) = 3 (x + 1) \ln (2) \Rightarrow \ln (a) = 3 \ln (2^{x + 1}) (1)

\log_{a} y = \frac{1}{3} \Rightarrow \ln (a) = 3 \ln (y) (2)

(1) = (2) \Rightarrow \ln (y) = \ln (2^{x + 1}) \Rightarrow y = 2^{x + 1} ✓

Answered by aba last updated on 11/Jun/23

\log_{a} (y) = \frac{1}{3} = \frac{\log_{2} (a)}{{3 \log}_{2} (a)} \Rightarrow \log_{a} (y) = \frac{\log_{8} (a)}{\log_{2} (a)}

\Rightarrow \log_{a} (y) = \frac{x + 1}{\log_{2} (a)} \Rightarrow \ln (y) = (x + 1) \ln (2)

\Rightarrow y = 2^{x + 1} ✓

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