Question

Expand log
root x cube / root Y square

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Answer 1

Answer:

After expansion we get   $\frac{3}{2}\times log\,x-log\,y$

Step-by-step explanation:

Given Expression:

$log\,(\frac{\sqrt{x^3}}{\sqrt{y^2}})$

We need to expand the given expression.

We use the following result,

log (a/b) = log a - log b

$log\,a^n=n.log\,a$

$(x^a)^b=x^{ab}$

Consider,

$log\,(\frac{\sqrt{x^3}}{\sqrt{y^2}})$

$=log\,\sqrt{x^3}-log\,\sqrt{y^2}$

$=log\,(x^3)^{\frac{1}{2}}-log\,(y^2)^{\frac{1}{2}}$

$=log\,(x)^{3\times\frac{1}{2}}-log\,(y)^{2\times\frac{1}{2}}$

$=log\,(x)^{\frac{3}{2}}-log\,(y)^{1}$

$=\frac{3}{2}\times log\,x-log\,y$

Therefore, After expansion we get   $\frac{3}{2}\times log\,x-log\,y$

Answer 2

The expanded form of given expression is $\dfrac{3}{2}\log x-\log y$.

Step-by-step explanation:

Consider the given expression is

$\log (\dfrac{\sqrt{x^3}}{\sqrt{y^2}})$

Using the properties of exponents it can be written as

$\log (\dfrac{(x^3)^{1/2}}{(y^2)^{1/2}})$            $[\because \sqrt{x}=x^{1/2}]$

$\log (\dfrac{x^{3/2}}{y})$             $[\because (x^m)^n=x^{mn}]$

Using properties of log we get

$\log x^{3/2}-\log y$             $[\because \log (\dfrac{a}{b})=\log a-\log b]$

$\dfrac{3}{2}\log x-\log y$           $[\because \log (a^b)=b\log a]$

Therefore, the expanded form of given expression is $\dfrac{3}{2}\log x-\log y$.

#Learn more

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