Express in terms of sums and differences of logarithms ln 9/8x^8 y

ANSWER

[tex]ln( \frac{9}{8 {x}^{8}y } ) =2ln( {3}) -( 3 ln( {2}) +8 ln( {x}) + ln( y ) )[/tex]

EXPLANATION

The given logarithmic expression is:

[tex] ln( \frac{9}{8 {x}^{8}y } ) [/tex]

Recall and apply the quotient rule:

[tex] ln( \frac{a}{b} ) = ln(a) - ln(b) [/tex]

This gives;

[tex] ln( \frac{9}{8 {x}^{8}y } ) = ln( 9 ) - ln( 8 {x}^{8}y ) [/tex]

Use the product rule:

[tex] ln(ab) = ln(a) + ln(b) [/tex]

[tex]ln( \frac{9}{8 {x}^{8}y } ) = ln( 9 ) -( ln( 8 ) + ln( {x}^{8}) + ln( y ) )[/tex]

[tex]ln( \frac{9}{8 {x}^{8}y } ) = ln( {3}^{2} ) -( ln( {2}^{3} ) + ln( {x}^{8}) + ln( y ) )[/tex]

Apply the power rule:

[tex] ln( {a}^{k} ) = k \: ln(a) [/tex]

[tex]ln( \frac{9}{8 {x}^{8}y } ) =2ln( {3}) -( 3 ln( {2}) +8 ln( {x}) + ln( y ) )[/tex]