Question

Expand the following logarithm:
$log_{5}( \frac{ {x}^{2} }{y} )^{6}$
​

Answer 1

Answer:

$\displaystyle 12 \log_{5}( {x}^{} {)}^{} - 6 \log_{5}(y ^{} )$

Step-by-step explanation:

we would like to expand the following

$\displaystyle \log_{5}\bigg( \frac{{x}^{2}}{y} \bigg) ^{6}$

since we have a division of two different variable we can consider using division logarithm rule

$\displaystyle \log_{5}( {x}^{2} {)}^{6} - \log_{5}(y) ^{6}$

use law of exponent:

$\displaystyle \log_{5}( {x}^{12} {)}^{} - \log_{5}(y ^{6} )$

by exponent logarithm rule we acquire:

$\displaystyle 12 \log_{5}( {x}^{} {)}^{} - 6 \log_{5}(y ^{} )$

and we are done!