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Question Number 195708 by mokys last updated on 08/Aug/23

$$\mathit{P}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{t}\mathit{h}\mathit{a}\mathit{t}:{\mathit{l}\mathit{o}\mathit{g}}_{(\sqrt{\mathit{a}}-\mathit{b})}(\sqrt{\mathit{a}}+\mathit{b})=-1$$

Answered by JDamian last updated on 08/Aug/23

$$b=0\Rightarrow {\log }_{\sqrt{a}}\left(\sqrt{a}\right)=1\ne -1$$

Answered by Frix last updated on 08/Aug/23

$$\frac{\ln (\sqrt{a}+b)}{\ln (\sqrt{a}-b)}=-1$$$$\mathrm{Defined}\mathrm{for}a⩾0\wedge \sqrt{a}>b\wedge a\ne {(b+1)}^2$$$$\mathrm{True}\mathrm{only}\mathrm{if}a=b^2+1$$

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