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What is the value of the logarithmic expression?

[tex]\log_5\left(\frac{1}{25}\right)[/tex]

Enter your answer in the box.

[tex]\log_5\left(\frac{1}{25}\right) =[/tex] [tex]\square[/tex]

Answer :

To solve [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex], we can use properties of logarithms and exponents:

1. Recognize the Fraction as a Negative Exponent:
[tex]\[
\frac{1}{25} = 25^{-1}
\][/tex]

2. Rewrite Using Logarithmic Properties:
The property [tex]\(\log_b(a^{-1}) = -\log_b(a)\)[/tex] allows us to write:
[tex]\[
\log_5\left(\frac{1}{25}\right) = -\log_5(25)
\][/tex]

3. Express the Number as a Power of the Base:
Notice that [tex]\(25\)[/tex] can be expressed as a power of [tex]\(5\)[/tex], since [tex]\(25 = 5^2\)[/tex].

4. Apply Another Logarithmic Property:
The property [tex]\(\log_b(b^c) = c\)[/tex] tells us that:
[tex]\[
\log_5(25) = \log_5(5^2) = 2\log_5(5)
\][/tex]

Since [tex]\(\log_5(5) = 1\)[/tex] (because any base raised to the power 1 is itself), we have:
[tex]\[
\log_5(25) = 2 \times 1 = 2
\][/tex]

5. Combine the Steps:
Now substitute back to find the original expression:
[tex]\[
\log_5\left(\frac{1}{25}\right) = -\log_5(25) = -2
\][/tex]

Therefore, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].

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