resolve-logx-y-logy-x-x-y-y-x-

Question Number 78314 by mfwajoel1 last updated on 15/Jan/20

r e s o l v e

{_{{l o g x}_{y} = {l o g y}_{x}}^{x^{y} = y^{x}}

Commented by john santu last updated on 16/Jan/20

w h a t t h e m e a n i n g \log y_{x} ?

Answered by MJS last updated on 16/Jan/20

if you mean \log_{y} x = \log_{x} y

(1) \Rightarrow y \ln x = x \ln y

(2) \Rightarrow \frac{\ln x}{\ln y} = \frac{\ln y}{\ln x} \Rightarrow \ln y = \pm \ln x \Rightarrow y = x \lor y = \frac{1}{x}

[\frac{a}{b} = \frac{b}{a} \Rightarrow a^{2} = b^{2} \Rightarrow b = \pm a]

case 1 y = x

\Rightarrow (1) x^{x} = x^{x} true for x \in C ∖ {0}

case 2 y = \frac{1}{x}

\Rightarrow (1) \frac{\ln x}{x} = x \ln \frac{1}{x}

\frac{\ln x}{x} = - x \ln x

\frac{1}{x} = - x \Rightarrow x = \pm i \Rightarrow y = \mp i

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