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Question Number 92727 by I want to learn more last updated on 08/May/20

Solve: x^y = y^x ....... (i) 3^x = 15^y ...... (ii) x ≠ y, x, y ∈ R

$$\mathrm{Solve}:\:\:\:\:\:\mathrm{x}^{\mathrm{y}} \:\:=\:\:\mathrm{y}^{\mathrm{x}} \:\:\:\:\:.......\:\:\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}^{\mathrm{x}} \:\:=\:\:\mathrm{15}^{\mathrm{y}} \:\:\:\:......\:\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\mathrm{x}\:\:\neq\:\:\mathrm{y},\:\:\:\:\:\:\:\mathrm{x},\:\:\mathrm{y}\:\in\:\mathbb{R} \\ $$

Answered by mr W last updated on 08/May/20

x ln 3=y ln 15 ⇒(x/y)=((ln 15)/(ln 3))=1+((ln 5)/(ln 3))=k y ln x=x ln y ⇒(x/y)=((ln x)/(ln y)) k=((ln k+ln y)/(ln y))=((ln k)/(ln y))+1 ⇒ln y=((ln k)/(k−1)) ⇒y=e^((ln k)/(k−1)) =e^((ln (1+((ln 5)/(ln 3))))/((ln 5)/(ln 3))) =e^((ln 3 ln (1+((ln 5)/(ln 3))))/(ln 5)) =1.8512 ⇒x=ky=(1+((ln 5)/(ln 3)))e^((ln 3 ln (1+((ln 5)/(ln 3))))/(ln 5)) =4.5632

$${x}\:\mathrm{ln}\:\mathrm{3}={y}\:\mathrm{ln}\:\mathrm{15} \\ $$$$\Rightarrow\frac{{x}}{{y}}=\frac{\mathrm{ln}\:\mathrm{15}}{\mathrm{ln}\:\mathrm{3}}=\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}={k} \\ $$$${y}\:\mathrm{ln}\:{x}={x}\:\mathrm{ln}\:{y} \\ $$$$\Rightarrow\frac{{x}}{{y}}=\frac{\mathrm{ln}\:{x}}{\mathrm{ln}\:{y}} \\ $$$${k}=\frac{\mathrm{ln}\:{k}+\mathrm{ln}\:{y}}{\mathrm{ln}\:{y}}=\frac{\mathrm{ln}\:{k}}{\mathrm{ln}\:{y}}+\mathrm{1} \\ $$$$\Rightarrow\mathrm{ln}\:{y}=\frac{\mathrm{ln}\:{k}}{{k}−\mathrm{1}} \\ $$$$\Rightarrow{y}={e}^{\frac{\mathrm{ln}\:{k}}{{k}−\mathrm{1}}} ={e}^{\frac{\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}\right)}{\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}}} ={e}^{\frac{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}\right)}{\mathrm{ln}\:\mathrm{5}}} =\mathrm{1}.\mathrm{8512} \\ $$$$\Rightarrow{x}={ky}=\left(\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}\right){e}^{\frac{\mathrm{ln}\:\mathrm{3}\:\mathrm{ln}\:\left(\mathrm{1}+\frac{\mathrm{ln}\:\mathrm{5}}{\mathrm{ln}\:\mathrm{3}}\right)}{\mathrm{ln}\:\mathrm{5}}} =\mathrm{4}.\mathrm{5632} \\ $$

Commented by I want to learn more last updated on 08/May/20

Thanks sir

$$\mathrm{Thanks}\:\mathrm{sir} \\ $$

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