Question

. If 3^x= 4^y = 12^z then the value of z is-
(a)xy/x+y (b)xy/x-y (c)x/x+y (d)y/x+y​

Answer 1

Let,

$\longrightarrow\sf{3^x=4^y=12^z=k}$

Then we have,

$\longrightarrow\sf{3^x=k\quad\implies\quad x=\dfrac{\log k}{\log 3}}$

$\longrightarrow\sf{4^y=k\quad\implies\quad y=\dfrac{\log k}{\log 4}}$

$\longrightarrow\sf{12^z=k\quad\implies\quad z=\dfrac{\log k}{\log 12}}$

Hence,

$\longrightarrow\sf{z=\dfrac{\log k}{\log 12}}$

$\longrightarrow\sf{z=\dfrac{\log k}{\log(3\times4)}}$

Since $\sf{\log(ab)=\log a+\log b,}$

$\longrightarrow\sf{z=\dfrac{\log k}{\log3+\log4}}$

$\longrightarrow\sf{z=\left[\dfrac{\log 3+\log4}{\log k}\right]^{-1}}$

$\longrightarrow\sf{z=\left[\dfrac{\log 3}{\log k}+\dfrac{\log4}{\log k}\right]^{-1}}$

$\longrightarrow\sf{z=\left[\dfrac{1}{x}+\dfrac{1}{y}\right]^{-1}}$

$\longrightarrow\sf{z=\left[\dfrac{x+y}{xy}\right]^{-1}}$

$\longrightarrow\sf{\underline{\underline{z=\dfrac{xy}{x+y}}}}$

Hence (a) is the answer.