Question

if 2^x=3^y=12^z, then show that 1/z=1/y+2/x​

Answer 1

GIVEN:-

• if $\rm{2^x=3^y=12^z}$

TO SHOW

• $\rm{\dfrac{1}{z}=\dfrac{1}{y}+\dfrac{2}{x}}$.

Now,

$\implies\rm{2^x=3^y=12^z= k}$

$\implies\rm{2^x=k}$

$\implies\rm{2=k^\frac{1}{x}}$

$\implies\rm{3^y=k}$

$\implies\rm{3=k^\frac{1}{k}}$

$\implies\rm{12^z=k}$

$\implies\rm{12=k^\frac{1}{z}}$.

Now,

As we know,

$\implies\tt{12=2^2\times{3}}$

$\implies\tt{k^{\dfrac{1}{z}}=(k^{\dfrac{1}{x}})^2\times{k^\dfrac{1}{z}}$

$\implies\tt{k^\frac{2}{x}+^\frac{1}{y}=k^\frac{1}{z}}$.

Bases are same and equating the Powers

$\implies\mathcal{\dfrac{2}{x}+\dfrac{1}{x}=\dfrac{1}{z}}$.

Hence, Proved.