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Answer :
To solve the problem of finding [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex], let's break it down into simpler steps.
1. Understand the expression: [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] asks us to find the power to which the base 5 must be raised to result in [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite [tex]\(\frac{1}{25}\)[/tex]: First, note that [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex]. Therefore, [tex]\(\frac{1}{25} = \frac{1}{5^2} = 5^{-2}\)[/tex].
3. Apply the logarithm property: The logarithmic property states that [tex]\(\log_b(b^x) = x\)[/tex]. Applying this to our expression:
[tex]\[
\log_5\left(5^{-2}\right) = -2
\][/tex]
4. Result: Thus, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
5. Final answer: The solution rounded to one decimal place is [tex]\(-2.0\)[/tex].
This means that 5 raised to the power of [tex]\(-2\)[/tex] equals [tex]\(\frac{1}{25}\)[/tex], confirming that [tex]\(\log_5\left(\frac{1}{25}\right) = -2.0\)[/tex].
1. Understand the expression: [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] asks us to find the power to which the base 5 must be raised to result in [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite [tex]\(\frac{1}{25}\)[/tex]: First, note that [tex]\(25\)[/tex] is [tex]\(5^2\)[/tex]. Therefore, [tex]\(\frac{1}{25} = \frac{1}{5^2} = 5^{-2}\)[/tex].
3. Apply the logarithm property: The logarithmic property states that [tex]\(\log_b(b^x) = x\)[/tex]. Applying this to our expression:
[tex]\[
\log_5\left(5^{-2}\right) = -2
\][/tex]
4. Result: Thus, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
5. Final answer: The solution rounded to one decimal place is [tex]\(-2.0\)[/tex].
This means that 5 raised to the power of [tex]\(-2\)[/tex] equals [tex]\(\frac{1}{25}\)[/tex], confirming that [tex]\(\log_5\left(\frac{1}{25}\right) = -2.0\)[/tex].
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