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Question Number 36450 by Raj Singh last updated on 02/Jun/18

Commented by tanmay.chaudhury50@gmail.com last updated on 02/Jun/18

Commented by ajfour last updated on 03/Jun/18

right!

$${right}! \\ $$

Answered by ajfour last updated on 02/Jun/18

3^x =5^y =3^z (5^2 )^z ⇒ xln 3=yln 5=zln 3+2zln 5 ⇒ y=((xln 3)/(ln 5)) ....(i) z=((xln 3)/(ln 3+2ln 5))= ((x(((ln 3)/(ln 5))))/((((ln 3)/(ln 5)))+2)) using (i) z = (y/((y/x)+2)) = ((xy)/(2x+y)) .

$$\mathrm{3}^{{x}} =\mathrm{5}^{{y}} =\mathrm{3}^{{z}} \left(\mathrm{5}^{\mathrm{2}} \right)^{{z}} \\ $$$$\Rightarrow\:\:{x}\mathrm{ln}\:\mathrm{3}={y}\mathrm{ln}\:\mathrm{5}={z}\mathrm{ln}\:\mathrm{3}+\mathrm{2}{z}\mathrm{ln}\:\mathrm{5} \\ $$$$\Rightarrow\:\:{y}=\frac{{x}\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\:\:\:\:\:\:\:\:....\left({i}\right) \\ $$$$\:\:\:\:\:\:{z}=\frac{{x}\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{3}+\mathrm{2ln}\:\mathrm{5}}=\:\frac{{x}\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\right)}{\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\right)+\mathrm{2}} \\ $$$${using}\:\left({i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{z}\:=\:\frac{{y}}{\frac{{y}}{{x}}+\mathrm{2}}\:=\:\frac{{xy}}{\mathrm{2}{x}+{y}}\:. \\ $$

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