Answer:
[tex] log ( \frac { 5 x } { 4 y} ) \implies [/tex] [tex] log ( 5 ) + log ( x ) - log ( 4 ) + log ( y ) [/tex]
Step-by-step explanation:
We are given the following for which we are to expand the logarithm:
[tex] log ( \frac { 5 x } { 4 y} ) [/tex]
Expanding the log by applying the rules of expanding the logarithms by changing the division into subtraction:
[tex] log ( 5 x ) - log ( 4 y ) [/tex]
[tex] log ( 5 ) + log ( x ) - log ( 4 ) + log ( y ) [/tex]
[tex]\bf \begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( \frac{x}{y}\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \log\left( \cfrac{5x}{4y} \right)\implies \log(5x)-\log(4y)\implies [\log(5)+\log(x)]-[\log(4)+\log(y)] \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill \log(5)+\log(x)-\log(4)-\log(y)~\hfill[/tex]
