Question

prove: \\(\log_a \left(\frac{x}{y}\
ight) = \log_a x - \log_a y\\).
let \\(\log_a x = m\\) and \\(\log_a y = n\\). then by the definition of logarithms, \\(x = \square\\) and \\(y = \square\\).

Explanation:

Step1: Recall logarithm definition

The definition of a logarithm states that if $\log_{a}b = c$, then $b = a^{c}$.

Step2: Apply to $\log_{a}x = m$

Using the definition, for $\log_{a}x = m$, we have $x = a^{m}$.

Step3: Apply to $\log_{a}y = n$

Similarly, for $\log_{a}y = n$, we have $y = a^{n}$.

Answer:

$x = \boldsymbol{a^{m}}$ and $y = \boldsymbol{a^{n}}$