x-12-log-3-x-x-18-log-2-x-find-x-

Question Number 138690 by KwesiDerek last updated on 16/Apr/21

{(\frac{x}{12})}^{\log_{\sqrt{3}} x} = {(\frac{x}{18})}^{\log_{\sqrt{2}} x}

find x

Commented by liberty last updated on 16/Apr/21

x = 36 \Rightarrow {(\frac{36}{12})}^{\log_{\sqrt{3}} (36)} = 3^{\log_{3} {(36)}^{2}} = 36^{2}

\Rightarrow {(\frac{36}{18})}^{\log_{\sqrt{2}} (36)} = {(2)}^{\log_{2} {(36)}^{2}} = 36^{2}

Answered by liberty last updated on 16/Apr/21

x^{\log_{\sqrt{3}} (\frac{x}{12})} = x^{\log_{\sqrt{2}} (\frac{x}{18})}

\Rightarrow \log_{3} (\frac{x}{12}) = \log_{2} (\frac{x}{18})

\Rightarrow \frac{\ln x - \ln 12}{\ln 3} = \frac{\ln x - \ln 18}{\ln 2}

\Rightarrow \ln 2 (\ln x - \ln 12) = \ln 3 (\ln x - \ln 18)

\Rightarrow \ln x (\ln 2 - \ln 3) = \ln 2 \ln 12 - \ln 3 \ln 18

\Rightarrow \ln x = \frac{\ln 2 (2 \ln 2 + \ln 3) - \ln 3 (2 \ln 3 + \ln 2)}{\ln 2 - \ln 3}

\ln x = \frac{2 \ln^{2} 2 + \ln 2 \ln 3 - 2 \ln^{2} 3 - \ln 2 \ln 3}{\ln 2 - \ln 3}

\ln x = \frac{2 (\ln 2 - \ln 3) (\ln 2 + \ln 3)}{\ln 2 - \ln 3}

\Leftrightarrow \ln x = \ln 36; x = 36

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