How do you express a difference of logarithm log_a *(x/y)?

Jun 19, 2015

${\log}_{a} \left(\frac{x}{y}\right) = {\log}_{a} x - {\log}_{a} y$

Explanation:

This is gotten from the result ${\log}_{b} \left(\frac{A}{B}\right) = {\log}_{b} A - {\log}_{b} B$

If you're curious as to how this is possible then continue reading

Suppose, ${b}^{m} = A \text{ }$ and $\text{ } {b}^{n} = B$

$\implies {\log}_{b} A = m \text{ }$ and $\text{ } {\log}_{b} B = n$

From the first statement, $\frac{A}{B} = {b}^{m} / {b}^{n}$

And by law of indices,

$\implies \frac{A}{B} = {b}^{m - n}$
$\implies {\log}_{b} \left(\frac{A}{B}\right) = m - n$

Recall that, $\implies {\log}_{b} A = m \text{ }$ and $\text{ } {\log}_{b} B = n$

$\implies \textcolor{b l u e}{{\log}_{b} \left(\frac{A}{B}\right) = {\log}_{b} A - {\log}_{b} B}$