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The value of [tex]$\log \left(\frac{p}{q}\right)$[/tex] is

(a) [tex]$\log p - \log q$[/tex]

(b) [tex]$\frac{\log p}{\log q}$[/tex]

(c) [tex][tex]$\log p + \log q$[/tex][/tex]

(d) [tex]$\log q - \log p$[/tex]


Sagot :

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]