Question

if 2 to the power x=3 to the power y=12 to the power z ,then show that 1/z=1/y+2/x
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Answer 1

Solution:-

Given :-

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

To prove

$\to \rm \: \dfrac{1}{z} - \dfrac{1}{y} = \dfrac{2}{x}$

Now we can write as

$\rm \to \: \dfrac{1}{z} = \dfrac{1}{y} + \dfrac{2}{x}$

Let

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

Now we can say

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

Take

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

take log on both side

$\to\rm \: log2 {}^{x} = log \: k$

$\rm \to \: x log \: 2 = log \: k$

$\rm \: \to \: \dfrac{1}{x} = \dfrac{ log \: 2 }{ log \: k }$

By using logarithmic properties

$\to \rm \: \dfrac{1}{x} = log_{k}2$

Now take

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

We can write as

$\rm \: \to \: \dfrac{1}{y} = log_{k}3$

now take

$\rm12 {}^{z} = k$

We can write as

$\to \rm \dfrac{1}{z} = log_{k}12$

Take RHS

$\to \rm \dfrac{1}{z} = log_{k}12$

Using logarithmic properties

$\to \rm \dfrac{1}{z} = log_{k}( {2}^{2} \times 3)$

$\to \rm \dfrac{1}{z} = log_{k} 2 {}^{2} + log_{k}3$

$\to \rm \dfrac{1}{z} =2 log_{k} 2 {}^{} + log_{k}3$

$\boxed{ \rm \to \: \dfrac{1}{z} = \dfrac{2}{x} + \dfrac{1}{y} }$

Hence proved