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Question Number 85694 by john santu last updated on 24/Mar/20

$${\log }_{\left(\frac{x}{x-3}\right)}\left(7\right)<{\log }_{\left(\frac{x}{3}\right)}\left(7\right)$$

Commented by jagoll last updated on 24/Mar/20

$$\frac{1}{{\log }_7\left(\frac{\mathrm{x}}{\mathrm{x}-3}\right)}<\frac{1}{{\log }_7\left(\frac{\mathrm{x}}{3}\right)}$$$$\left(1\right)\frac{\mathrm{x}}{\mathrm{x}-3}>0\wedge \frac{\mathrm{x}}{3}>0$$$$\Rightarrow \mathrm{x}>3$$$$\left(2\right){\log }_7\left(\frac{\mathrm{x}}{3}\right)<{\log }_7\left(\frac{\mathrm{x}}{\mathrm{x}-3}\right)$$$$\frac{\mathrm{x}}{3}-\frac{\mathrm{x}}{\mathrm{x}-3}<0$$$$\frac{\mathrm{x}(\mathrm{x}-6)}{3(\mathrm{x}-3)}<0\Rightarrow \mathrm{x}<0\vee 3<\mathrm{x}<6$$$$\mathrm{the}\mathrm{solution}\Rightarrow 3<\mathrm{x}<6$$

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