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Answer :
To find the value of the logarithmic expression [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex], let's follow these steps:
1. Understand the Expression: We are looking for the power to which the base 5 must be raised to yield [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite the Fraction as an Exponent: Notice that [tex]\(\frac{1}{25}\)[/tex] can be rewritten in terms of a power of 5. We know that [tex]\(25 = 5^2\)[/tex]. Therefore, [tex]\(\frac{1}{25}\)[/tex] can be expressed as the reciprocal of 25:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
3. Apply the Properties of Logarithms: Now, we want to evaluate [tex]\(\log_5(5^{-2})\)[/tex]. One of the properties of logarithms states that [tex]\(\log_b(b^a) = a\)[/tex]. This means that the logarithm of a number where the base is raised to an exponent equals the exponent.
4. Calculate the Logarithm:
[tex]\[
\log_5(5^{-2}) = -2
\][/tex]
Therefore, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
1. Understand the Expression: We are looking for the power to which the base 5 must be raised to yield [tex]\(\frac{1}{25}\)[/tex].
2. Rewrite the Fraction as an Exponent: Notice that [tex]\(\frac{1}{25}\)[/tex] can be rewritten in terms of a power of 5. We know that [tex]\(25 = 5^2\)[/tex]. Therefore, [tex]\(\frac{1}{25}\)[/tex] can be expressed as the reciprocal of 25:
[tex]\[
\frac{1}{25} = \frac{1}{5^2} = 5^{-2}
\][/tex]
3. Apply the Properties of Logarithms: Now, we want to evaluate [tex]\(\log_5(5^{-2})\)[/tex]. One of the properties of logarithms states that [tex]\(\log_b(b^a) = a\)[/tex]. This means that the logarithm of a number where the base is raised to an exponent equals the exponent.
4. Calculate the Logarithm:
[tex]\[
\log_5(5^{-2}) = -2
\][/tex]
Therefore, the value of [tex]\(\log_5\left(\frac{1}{25}\right)\)[/tex] is [tex]\(-2\)[/tex].
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