To evaluate the expression [tex]\(\log_5 \frac{1}{25}\)[/tex], follow these steps:
1. Understand the expression:
The expression [tex]\(\log_5 \frac{1}{25}\)[/tex] is asking for the power to which the base 5 must be raised to result in [tex]\(\frac{1}{25}\)[/tex].
2. Express [tex]\(\frac{1}{25}\)[/tex] in terms of base 5:
We know that [tex]\(25\)[/tex] can be written as [tex]\(5^2\)[/tex]. Therefore:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
3. Rewrite the logarithm:
Using the above expression, we can rewrite [tex]\(\log_5 \frac{1}{25}\)[/tex] as [tex]\(\log_5 5^{-2}\)[/tex].
4. Use the logarithm power rule:
The logarithm power rule states that [tex]\(\log_b (a^c) = c \cdot \log_b a\)[/tex]. In this case:
[tex]\[
\log_5 5^{-2} = -2 \cdot \log_5 5
\][/tex]
5. Evaluate the logarithm of the same base:
The property of logarithms [tex]\(\log_b b = 1\)[/tex] gives us:
[tex]\[
\log_5 5 = 1
\][/tex]
So,
[tex]\[
-2 \cdot \log_5 5 = -2 \cdot 1 = -2
\][/tex]
Thus, the value of [tex]\(\log_5 \frac{1}{25}\)[/tex] is [tex]\(-2\)[/tex].
The correct answer is [tex]\(-2\)[/tex].
