Dolly Robinson

2021-08-17

Expand using logarithmic properties. Where possible, evaluate logarithmic expressions.

${\mathrm{log}}_{5}\left(\frac{{x}^{3}\sqrt{y}}{125}\right)$

Alix Ortiz

Skilled2021-08-18Added 109 answers

Step 1

On simplification, we get

${\mathrm{log}}_{5}\left(\frac{{x}^{3}\sqrt{y}}{125}\right)={\mathrm{log}}_{5}\left(125\right)\text{}[\because {\mathrm{log}}_{a}\left(\frac{m}{n}\right)={\mathrm{log}}_{a}\left(m\right)-{\mathrm{log}}_{a}\left(n\right)]$

$={\mathrm{log}}_{5}\left({x}^{3}\right)+{\mathrm{log}}_{5}\left(\sqrt{y}\right)-{\mathrm{log}}_{5}\left({5}^{3}\right)\text{}[\because {\mathrm{log}}_{a}\left(mn\right)={\mathrm{log}}_{a}\left(m\right)+{\mathrm{log}}_{a}\left(n\right)]$

$={\mathrm{log}}_{5}\left({x}^{3}\right)+{\mathrm{log}}_{5}\left({y}^{\frac{1}{2}}\right)-{\mathrm{log}}_{5}\left({5}^{3}\right)$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3{\mathrm{log}}_{5}\left(5\right)\text{}[\because {\mathrm{log}}_{a}\left({m}^{n}\right)=n{\mathrm{log}}_{a}\left(m\right)]$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3\left(1\right)\text{}[\because {\mathrm{log}}_{a}\left(a\right)=1]$

$=3{\mathrm{log}}_{5}\left(x\right)+\frac{1}{2}{\mathrm{log}}_{5}\left(y\right)-3$

On simplification, we get