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Answer :
To solve the problem of finding the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex], you can follow these steps:
1. Understand the Expression:
- You need to find the logarithm of [tex]\(\frac{1}{25}\)[/tex] with base 5.
2. Rewrite [tex]\(\frac{1}{25}\)[/tex]:
- Notice that [tex]\(25\)[/tex] is a power of [tex]\(5\)[/tex], specifically [tex]\(25 = 5^2\)[/tex].
- Thus, [tex]\(\frac{1}{25}\)[/tex] can be rewritten as [tex]\(5^{-2}\)[/tex].
3. Apply the Logarithm Power Rule:
- The power rule of logarithms states that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex].
- Apply this rule to [tex]\(\log_5(5^{-2})\)[/tex]:
[tex]\[
\log_5(5^{-2}) = -2 \cdot \log_5(5)
\][/tex]
4. Evaluate [tex]\(\log_5(5)\)[/tex]:
- By definition, [tex]\(\log_5(5) = 1\)[/tex] because the logarithm of a number with its own base is 1.
5. Compute the Final Answer:
- Substitute back into the equation:
[tex]\[
-2 \cdot 1 = -2
\][/tex]
Therefore, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
1. Understand the Expression:
- You need to find the logarithm of [tex]\(\frac{1}{25}\)[/tex] with base 5.
2. Rewrite [tex]\(\frac{1}{25}\)[/tex]:
- Notice that [tex]\(25\)[/tex] is a power of [tex]\(5\)[/tex], specifically [tex]\(25 = 5^2\)[/tex].
- Thus, [tex]\(\frac{1}{25}\)[/tex] can be rewritten as [tex]\(5^{-2}\)[/tex].
3. Apply the Logarithm Power Rule:
- The power rule of logarithms states that [tex]\(\log_b(a^c) = c \cdot \log_b(a)\)[/tex].
- Apply this rule to [tex]\(\log_5(5^{-2})\)[/tex]:
[tex]\[
\log_5(5^{-2}) = -2 \cdot \log_5(5)
\][/tex]
4. Evaluate [tex]\(\log_5(5)\)[/tex]:
- By definition, [tex]\(\log_5(5) = 1\)[/tex] because the logarithm of a number with its own base is 1.
5. Compute the Final Answer:
- Substitute back into the equation:
[tex]\[
-2 \cdot 1 = -2
\][/tex]
Therefore, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
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