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Question Number 187672 by Humble last updated on 20/Feb/23

$$4^{-\frac{1}{x}}+6^{-\frac{1}{x}}=9^{-\frac{1}{x}}$$$$x\in \mathbb{R}$$

Answered by SEKRET last updated on 20/Feb/23

$$\frac{4^{-\frac{1}{\mathbf{x}}}}{9^{-\frac{1}{\mathbf{x}}}}+\frac{6^{\frac{-1}{\mathbf{x}}}}{9^{\frac{-1}{\mathbf{x}}}}=1$$$${\left(\frac{4}{9}\right)}^{\frac{-1}{\mathbf{x}}}+{\left(\frac{6}{9}\right)}^{\frac{-1}{\mathbf{x}}}=1$$$${\left(\frac{2}{3}\right)}^{2\cdot \frac{-1}{\mathbf{x}}}+{\left(\frac{2}{3}\right)}^{\frac{-1}{\mathbf{x}}}=1$$$${\mathbf{t}}^2+\mathbf{t}=1$$$${\mathbf{t}}^2+2\cdot \frac{1}{2}\cdot \mathbf{t}+\frac{1}{4}=1+\frac{1}{4}$$$${(\mathbf{t}+\frac{1}{2})}^2=\frac{5}{4}$$$$\mathbf{t}=\frac{-1\mp \sqrt{5}}{2}$$$${\left(\frac{2}{3}\right)}^{\frac{-1}{\mathbf{x}}}=\frac{\sqrt{5}-1}{2}$$$$\frac{-1}{\mathbf{x}}={\mathbf{log}}_{\left(\frac{2}{3}\right)}\left(\frac{\sqrt{5}-1}{2}\right)$$$$\frac{1}{\mathbf{x}}={\mathbf{log}}_{\left(\frac{3}{2}\right)}\left(\frac{\sqrt{5}-1}{2}\right)$$$$\mathbf{x}={\mathbf{log}}_{\left(\frac{\sqrt{5}-1}{2}\right)}\left(\frac{3}{2}\right)$$

Commented by Humble last updated on 20/Feb/23

$$nicesolution,sir$$

Answered by Rasheed.Sindhi last updated on 20/Feb/23

$$4^{-\frac{1}{x}}+6^{-\frac{1}{x}}=9^{-\frac{1}{x}}$$$${\left(\frac{4}{6}\right)}^{-1/x}+1={\left(\frac{9}{6}\right)}^{-1/x}$$$${\left(\frac{3}{2}\right)}^{1/x}+1={\left(\frac{2}{3}\right)}^{1/x}$$$$y+1=\frac{1}{y};y⩾0$$$$y^2+y-1=0$$$$y=\frac{-1+\sqrt{1+4}}{2}$$$$\log {\left(\frac{3}{2}\right)}^{1/x}=\log \left(\frac{-1+\sqrt{5}}{2}\right)$$$$\left(\frac{1}{x}\right)\log \left(\frac{3}{2}\right)=\log (-1+\sqrt{5})-\log 2$$$$x=\frac{\log 3-\log 2}{\log (-1+\sqrt{5})-\log 2}$$$$$$

Commented by Humble last updated on 20/Feb/23

$$Excellent!!$$

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