resolve {_(logx_y =logy_x ) ^(x^y =y^x )


Question and Answers Forum

All Questions Topic List
Others Questions
Previous in All Question Next in All Question
Previous in Others Next in Others
Question Number 78314 by mfwajoel1 last updated on 15/Jan/20
$resolve {_(logx_y =logy_x ) ^(x^y =y^x )$
$r e s o l v e$ ${_{{l o g x}_{y} = {l o g y}_{x}}^{x^{y} = y^{x}}$
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
	$if you mean log_y x =log_x y (1) ⇒ yln x =xln y (2) ⇒ ((ln x)/(ln y))=((ln y)/(ln x)) ⇒ ln y =±ln x ⇒ y=x∨y=(1/x) [(a/b)=(b/a) ⇒ a^2 =b^2 ⇒ b=±a] case 1 y=x ⇒ (1) x^x =x^x true for x∈C\{0} case 2 y=(1/x) ⇒ (1) ((ln x)/x)=xln (1/x) ((ln x)/x)=−xln x (1/x)=−x ⇒ x=±i ⇒ y=∓i$
	$if you mean \log_{y} x = \log_{x} y$ $(1) \Rightarrow y \ln x = x \ln y$ $(2) \Rightarrow \frac{\ln x}{\ln y} = \frac{\ln y}{\ln x} \Rightarrow \ln y = \pm \ln x \Rightarrow y = x \lor y = \frac{1}{x}$ $[\frac{a}{b} = \frac{b}{a} \Rightarrow a^{2} = b^{2} \Rightarrow b = \pm a]$ $case 1 y = x$ $\Rightarrow (1) x^{x} = x^{x} true for x \in C ∖ {0}$ $case 2 y = \frac{1}{x}$ $\Rightarrow (1) \frac{\ln x}{x} = x \ln \frac{1}{x}$ $\frac{\ln x}{x} = - x \ln x$ $\frac{1}{x} = - x \Rightarrow x = \pm i \Rightarrow y = \mp i$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum