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Question Number 23223 by abwayh last updated on 27/Oct/17

Answered by mrW1 last updated on 27/Oct/17

$$\Rightarrow \frac{\ln \mathrm{a}}{\ln \mathrm{x}}+\frac{\ln \mathrm{a}}{\ln \mathrm{y}}=4\times \frac{\ln \mathrm{a}}{\ln \left(\mathrm{xy}\right)}$$$$\Rightarrow \frac{1}{\ln \mathrm{x}}+\frac{1}{\ln \mathrm{y}}=\frac{4}{\ln \left(\mathrm{xy}\right)}$$$$\Rightarrow \frac{\ln \mathrm{x}+\ln \mathrm{y}}{\left(\ln \mathrm{x}\right)\left(\ln \mathrm{y}\right)}=\frac{4}{\ln \mathrm{x}+\ln \mathrm{y}}$$$$\Rightarrow {(\ln \mathrm{x}+\ln \mathrm{y})}^2=4\left(\ln \mathrm{x}\right)\left(\ln \mathrm{y}\right)$$$$\Rightarrow {(\ln \mathrm{x}-\ln \mathrm{y})}^2=0$$$$\Rightarrow \ln \mathrm{x}=\ln \mathrm{y}$$$$\Rightarrow \mathrm{x}=\mathrm{y}$$

Commented by abwayh last updated on 28/Oct/17

$$\mathrm{thank}\mathrm{you}\mathrm{very}\mathrm{much},\mathrm{sir}$$

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