Question

if 2 power x=5 power y= 10 power z then prove that 1/x +1/y= 1/z please help me to solve this question

Answer 1

this is answer for question

Answer 2

Answer:

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

Step-by-step explanation:

Given: $2^x=5^y=10^z$

Prove: $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$

Proof:

Let $2^x=5^y=10^z=k$

$2^x=k$

Taking log both sides.

$\log 2^x=\log k$

$x\log 2=\log k$                $[\because \log a^x=x\log a]$

Isolate variable term.

$x=\frac{\log k}{\log 2}$

After reciprocal we get

$\frac{1}{x}=\frac{\log 2}{\log k}$

Similarly,

$\frac{1}{y}=\frac{\log 5}{\log k}$

$\frac{1}{z}=\frac{\log 10}{\log k}$

We need to prove,

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

$LHS=\frac{1}{x}+\frac{1}{y}$

$LHS=\frac{\log 2}{\log k}+\frac{\log 5}{\log k}$

$LHS=\frac{\log 2+\log 5}{\log k}$         $[\because \log a+\log b=\log ab]$

$LHS=\frac{\log 10}{\log k}$

$LHS=\frac{1}{z}$

$LHS=RHS$

Hence proved.