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

Question

Expand the following logarithm:

`log_(5)( \frac{ {x}^(2) }{y} )^(6)`
​

Answer 1

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

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!