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Question Number 19382 by chernoaguero@gmail.com last updated on 10/Aug/17

$$\mathrm{If}{\log }_2(9^{x-1}+7)-{\log }_2(3^{x-1}+1)=2,$$$$\mathrm{then}\mathrm{the}\mathrm{values}\mathrm{of}x\mathrm{are}$$

Answered by khamizan833@yahoo.com last updated on 10/Aug/17

$$x=(1,2)$$

Answered by ajfour last updated on 10/Aug/17

$$\mathrm{let}{3}^{\mathbf{x}}=\mathrm{t}$$$$\Rightarrow \frac{\frac{{\mathrm{t}}^2}{9}+7}{\frac{\mathrm{t}}{3}+1}=4$$$${\mathrm{t}}^2+63=12\mathrm{t}+36$$$${\mathrm{t}}^2-12\mathrm{t}+27=0$$$$(\mathrm{t}-3)(\mathrm{t}-9)=0$$$$\Rightarrow \mathrm{t}=3^{\mathbf{x}}=3,9$$$$\Rightarrow \mathbf{x}=1\mathbf{or}\mathbf{x}=2.$$

Commented by chernoaguero@gmail.com last updated on 10/Aug/17

$$Thanksir$$

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