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 = a^m\\) and \\(y = a^n\\).
using the values found above for \\(x\\) and \\(y\\), what is \\(\frac{x}{y}\\) equal to?
\\(\frac{x}{y} = \square\\) (simplify your answer.)

Explanation:

Step1: Substitute x and y

We know \( x = a^m \) and \( y = a^n \). Substitute these into \( \frac{x}{y} \).
\( \frac{x}{y}=\frac{a^m}{a^n} \)

Step2: Apply exponent rule

Use the rule of exponents \( \frac{a^b}{a^c}=a^{b - c} \) (where \( a
eq0 \), \( b \) and \( c \) are real numbers). Here, \( b = m \) and \( c = n \), so \( \frac{a^m}{a^n}=a^{m - n} \).

Answer:

\( a^{m - n} \)