log-x-x-y-A-log-y-y-x-

Question Number 148573 by mathdanisur last updated on 29/Jul/21

{l o g}_{\sqrt{x}} \frac{x}{y} = A \Rightarrow {l o g}_{\sqrt{y}} \frac{y}{x} = ?

Answered by Ar Brandon last updated on 29/Jul/21

A = \log_{\sqrt{x}} \frac{x}{y} = \log_{\sqrt{x}} x - \log_{\sqrt{x}} y

= \log_{\sqrt{x}} \sqrt{x} \overset{2}{} - \frac{\log_{\sqrt{y}} y}{\log_{\sqrt{y}} \sqrt{x}} = 2 - \frac{2}{\log_{\sqrt{y}} \sqrt{x}}

\Rightarrow \log_{\sqrt{y}} \sqrt{x} = \frac{2}{2 - A}

\Rightarrow \log_{\sqrt{y}} \frac{y}{x} = 2 - \log_{\sqrt{y}} x = 2 - \frac{4}{2 - A}

Commented by puissant last updated on 29/Jul/21

\log (\frac{x}{y}) = \log (x) - \log (y) . .

Commented by Ar Brandon last updated on 29/Jul/21

Merci pour la remarque .

Commented by mathdanisur last updated on 29/Jul/21

T h a n k y o u S i r

Commented by Ar Brandon last updated on 29/Jul/21

{You}^{'} re welcome!

Answered by EDWIN88 last updated on 29/Jul/21

\log_{\sqrt{x}} (\frac{x}{y}) = A \Rightarrow 2 \log_{x} (\frac{x}{y}) = A

\log_{x} (\frac{y}{x}) = - \frac{A}{2} \Rightarrow \frac{y}{x} = x^{- \frac{A}{2}}

\Rightarrow y = x^{1 - \frac{A}{2}}

\log_{\sqrt{y}} (\frac{y}{x}) = 2 \log_{y} (\frac{y}{x})

= {2 \log}_{y} (x^{- \frac{A}{2}})

= - A \log_{y} (x) = - A \log_{(x^{1 - \frac{A}{2}})} (x)

= - A (\frac{2}{2 - A}) \log_{x} (x)

= \frac{2 A}{A - 2} .

Commented by mathdanisur last updated on 29/Jul/21

T h a n k y o u S e r