Question

if2^x=5^y=10^z then prove 1/x+1/y=1/z​

Answer 1

$\bold{ANSWER}$

$\rm{\frac{1}{x}+\frac{1}{y}=\frac{1}{z}}$

$\mathbb{EXPLANATION}$

$\rm{let\:2^x=5^y=10^z=k}$

$\rm{2^x=k}$....$\rm{Equation\:\:1}$

$\rm{5^y=k}$....$\rm{Equation\:\:2}$

$\rm{\left(10\right)^z=k}$....$\rm{Equation\:\:3}$

$\rm{2=k^{\frac{1}{x}}}.....\rm{Equation\:\:1}$

$\rm{5=k^{\frac{1}{y}}}....\rm{Equation\:\:2}$

$\rm{10=k^{\frac{1}{z}}}....\rm{Equation\:\:3}$

$\rm{10=2\times5}$

$\underline{Now\:put\:value\:of\:Equation\:1\:2\:And\:3\:in\:above\:Equation}$

$\rm{k^{\frac{1}{z}}=k^{\frac{1}{x}}\times\:k^{\frac{1}{y}}}$

$\rm{k^{\frac{1}{z}}=k^{\frac{1}{x}+\frac{1}{y}}}$

$\underline{Now\:compare\:powers\:of\:k\:we\:have}$

$\rm{\frac{1}{z}=\frac{1}{x}+\frac{1}{y}}$ $\bold{HENCE\:PROVED}$