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
$${right}! \\ $$
Answered by ajfour last updated on 02/Jun/18
$$\mathrm{3}^{{x}} =\mathrm{5}^{{y}} =\mathrm{3}^{{z}} \left(\mathrm{5}^{\mathrm{2}} \right)^{{z}} \\ $$$$\Rightarrow\:\:{x}\mathrm{ln}\:\mathrm{3}={y}\mathrm{ln}\:\mathrm{5}={z}\mathrm{ln}\:\mathrm{3}+\mathrm{2}{z}\mathrm{ln}\:\mathrm{5} \\ $$$$\Rightarrow\:\:{y}=\frac{{x}\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\:\:\:\:\:\:\:\:….\left({i}\right) \\ $$$$\:\:\:\:\:\:{z}=\frac{{x}\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{3}+\mathrm{2ln}\:\mathrm{5}}=\:\frac{{x}\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\right)}{\left(\frac{\mathrm{ln}\:\mathrm{3}}{\mathrm{ln}\:\mathrm{5}}\right)+\mathrm{2}} \\ $$$${using}\:\left({i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:{z}\:=\:\frac{{y}}{\frac{{y}}{{x}}+\mathrm{2}}\:=\:\frac{{xy}}{\mathrm{2}{x}+{y}}\:. \\ $$