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Answer :
To evaluate the logarithm [tex]\(\log_{25} \frac{1}{25}\)[/tex], let's break it down step by step:
1. Understand the Basic Definition of Logarithms:
The logarithm [tex]\(\log_b a = c\)[/tex] means [tex]\(b^c = a\)[/tex]. In this case, we need to find the exponent [tex]\(c\)[/tex] such that [tex]\(25^c = \frac{1}{25}\)[/tex].
2. Recognize the Given Expression:
We need [tex]\(\log_{25} \frac{1}{25}\)[/tex]. Here, the base [tex]\(b\)[/tex] is 25, and the result [tex]\(a\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
3. Identify the Relationship:
We know that [tex]\(\frac{1}{25}\)[/tex] can be rewritten as [tex]\(25^{-1}\)[/tex]. This is because taking the reciprocal of a number is the same as raising it to the power of [tex]\(-1\)[/tex].
4. Apply the Logarithm Definition:
We want to solve for [tex]\(c\)[/tex] in the equation [tex]\(25^c = 25^{-1}\)[/tex].
5. Determine the Exponent:
If [tex]\(25^c = 25^{-1}\)[/tex], it follows directly that [tex]\(c = -1\)[/tex].
Thus, the value of the logarithm [tex]\(\log_{25} \frac{1}{25}\)[/tex] is [tex]\(-1.0\)[/tex].
1. Understand the Basic Definition of Logarithms:
The logarithm [tex]\(\log_b a = c\)[/tex] means [tex]\(b^c = a\)[/tex]. In this case, we need to find the exponent [tex]\(c\)[/tex] such that [tex]\(25^c = \frac{1}{25}\)[/tex].
2. Recognize the Given Expression:
We need [tex]\(\log_{25} \frac{1}{25}\)[/tex]. Here, the base [tex]\(b\)[/tex] is 25, and the result [tex]\(a\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
3. Identify the Relationship:
We know that [tex]\(\frac{1}{25}\)[/tex] can be rewritten as [tex]\(25^{-1}\)[/tex]. This is because taking the reciprocal of a number is the same as raising it to the power of [tex]\(-1\)[/tex].
4. Apply the Logarithm Definition:
We want to solve for [tex]\(c\)[/tex] in the equation [tex]\(25^c = 25^{-1}\)[/tex].
5. Determine the Exponent:
If [tex]\(25^c = 25^{-1}\)[/tex], it follows directly that [tex]\(c = -1\)[/tex].
Thus, the value of the logarithm [tex]\(\log_{25} \frac{1}{25}\)[/tex] is [tex]\(-1.0\)[/tex].
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