Question

maths
${2}^{ x} = {3}^{y} = {12}^{z} .show \: that \: \frac{1}{z} = \frac{1}{y} + \frac{2}{x} .$

Answer 1

Hey!

Given :-
${2}^{x} = {3}^{y} = {12}^{z}$

Let ,
${2}^{x} = {3}^{y} = {12}^{z} = k(say)$

${2}^{x} = k \\ taking \: log \: at \: both \: side \: \\ log_{2}(k) = x$

Similarly ;

$log_{3}(k) = y \\ and \: \: log_{12}(k) = z$

Now ,
Taking LHS ;

$\frac{1}{z} = \frac{1}{ log_{12}(k) }$

$as > \frac{1}{ log_{a}(b) } = log_{b}(a)$
...( base change rule )

$\frac{1}{z} = log_{k}(12)$

Taking RHS ;

$\frac{1}{y} + \frac{2}{x} = \frac{1}{ log_{3}(k) } + \frac{2}{ log_{2}(k) }$

Using the same property [ base change rule ]

$\frac{1}{y} + \frac{2}{x} = log_{k}(3) + 2 log_{k}(2)$

$(m log_{a}(b) = log_{a}( {b}^{m} )$

So,

$\frac{1}{y} + \frac{2}{x} = log_{k}(3) + log_{k}(4)$

$( log_{a}(b) + log_{a}(c) = log_{a}(bc) )$

$\frac{1}{y} + \frac{2}{x} = log_{k}(12)$

Hence , LHS = RHS !!

proved !!

________________


${2}^{x} = {3}^{y} = {12}^{z} = k(say)$


[ Refer attachment ]

• Used properties :-

° When base are same while multiplying then power are added !!

a^m × a^n = a^(m+n)

° When base are equal then power are also equal.

a^m = a^n

Then , m = n !!