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Question Number 113996 by Aina Samuel Temidayo last updated on 16/Sep/20

Commented by MJS_new last updated on 16/Sep/20

$$\frac{\ln \frac{81}{8}}{2\ln \frac{9}{2}}=\frac{\ln \sqrt{\frac{81}{8}}}{\ln \frac{9}{2}}=\frac{\ln \frac{9}{\sqrt{8}}}{\ln \frac{9}{2}}={\log }_{\frac{9}{2}}\frac{9}{\sqrt{8}}$$

Answered by bobhans last updated on 17/Sep/20

$$9^x+\frac{1}{3}.9^x=2\sqrt{2}.2^x+\sqrt{2}.2^x$$$$\Rightarrow \frac{4}{3}.9^x=3\sqrt{2}.2^x$$$$\Rightarrow 9^{x-1}=2^{x-\frac{3}{2}};(x-1)\ln \left(9\right)=(x-\frac{3}{2})\ln \left(2\right)$$$$x\ln 9-x\ln 2=\ln 9-\frac{3}{2}\ln 2$$$$x.\ln \left(\frac{9}{2}\right)=\ln \left(\frac{9}{2\sqrt{2}}\right)\Rightarrow x=\log {}_{\frac{9}{2}}\left(\frac{9}{\sqrt{8}}\right)$$

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