Question

Expand the logarithm. log 5x/4y

Answer 1

`\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( (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`

Answer 2

`log ( \frac { 5 x } { 4 y} ) \implies`
`log ( 5 ) + log ( x ) - log ( 4 ) + log ( y )`

Explanation:

We are given the following for which we are to expand the logarithm:

`log ( \frac { 5 x } { 4 y} )`

Expanding the log by applying the rules of expanding the logarithms by changing the division into subtraction:

`log ( 5 x ) - log ( 4 y )`

`log ( 5 ) + log ( x ) - log ( 4 ) + log ( y )`