Certainly! We need to use the quotient property of logarithms to rewrite the given logarithmic expression [tex]\(\log_{12}\left(\frac{m}{n}\right)\)[/tex] as a difference of logarithms. Here's a detailed, step-by-step solution:
1. Identify the Quotient Property of Logarithms:
The quotient property of logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. Mathematically, this is written as:
[tex]\[
\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)
\][/tex]
where [tex]\(b\)[/tex] is the base of the logarithm, and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are positive real numbers.
2. Apply the Quotient Property:
Given the expression [tex]\(\log_{12}\left(\frac{m}{n}\right)\)[/tex], we apply the quotient property with base 12:
[tex]\[
\log_{12}\left(\frac{m}{n}\right) = \log_{12}(m) - \log_{12}(n)
\][/tex]
3. Simplify the Expression:
There are no further simplifications possible for this expression beyond applying the quotient property.
Thus, the logarithm [tex]\(\log_{12}\left(\frac{m}{n}\right)\)[/tex] can be written as the difference of logarithms:
[tex]\[
\log_{12}\left(\frac{m}{n}\right) = \log_{12}(m) - \log_{12}(n)
\][/tex]
### Domain
The domain of the logarithmic functions involved must be considered. For a logarithm [tex]\(\log_b(x)\)[/tex] to be defined, the argument [tex]\(x\)[/tex] must be a positive real number. Therefore, both [tex]\(m\)[/tex] and [tex]\(n\)[/tex] must be positive:
[tex]\[
m > 0 \quad \text{and} \quad n > 0
\][/tex]
We also note that since [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are in a fraction, [tex]\(n \neq 0\)[/tex] to avoid division by zero, but since [tex]\(n > 0\)[/tex], we have already satisfied this condition.
### Summary
The final result is:
[tex]\[
\log_{12}\left(\frac{m}{n}\right) = \log_{12}(m) - \log_{12}(n)
\][/tex]
The domain for this expression is:
[tex]\[
m > 0, \quad n > 0
\][/tex]
