In order to determine the value of [tex]\(\log \left(\frac{p}{q}\right)\)[/tex], we can use the properties of logarithms. One useful property of logarithms is that the logarithm of a quotient can be expressed as the difference of the logarithms. Specifically, the property states:
[tex]\[
\log \left(\frac{p}{q}\right) = \log p - \log q
\][/tex]
This property can be derived from the rules of exponents as follows:
1. Recall that the logarithm of a quotient can be written as the difference between two logarithms if both the numerator and denominator are positive.
2. Given that [tex]\(\log(p)\)[/tex] and [tex]\(\log(q)\)[/tex] are both defined (assuming [tex]\(p > 0\)[/tex] and [tex]\(q > 0\)[/tex]), we apply the quotient rule of logarithms:
[tex]\[
\log\left(\frac{p}{q}\right) = \log(p) - \log(q)
\][/tex]
Therefore, the value of [tex]\(\log \left(\frac{p}{q}\right)\)[/tex] is indeed:
[tex]\[
\boxed{\log p - \log q}
\][/tex]
Thus, the correct answer is:
(a) [tex]\(\log p - \log q\)[/tex]
