Question

Evaluate : log(root(x)) / log(x^2) *

Answer 1

Given:-

• $\sf \: \dfrac{log (\sqrt{x}) }{log( {x}^{2}) }$

We have to evaluate this

Answer:-

$\sf \: \dfrac{log (\sqrt{x}) }{log( {x}^{2}) }$

Using the formula,

${\blue{\bigstar}} \boxed{\sf{ \dfrac{log(a)}{log(b)} = log_b(a)}}$

$\longrightarrow \sf log_{x^2}(\sqrt{x})$

We know that, $\sf \sqrt{x} = x^{ { }^{1} \! / { }_{2}}$

$\longrightarrow \sf log_{x^2}(x^{ { }^{1} \! / { }_{2}})$

${\green{\bigstar}} \boxed{\sf{log_{a^m}(b^n) = \dfrac{n}{m}log_a(b)}}$

$\longrightarrow \sf \dfrac{1}{2}\times \dfrac{1}{2} log_{x}(x)$

We know that,

${\red{\bigstar}} \boxed{\sf{log_x(x) = 1 }}$

So,

$\longrightarrow \sf \dfrac{1}{4}\times 1$

$\longrightarrow \sf \dfrac{1}{4}$

Hence,

$\longrightarrow \underline{\underline{\sf{\green{ \dfrac{log (\sqrt{x}) }{log( {x}^{2}) } = \dfrac{1}{4} }}}}$