Recognize base and argument: Recognize the base of the logarithm and the argument. The base of the logarithm is 25, and the argument is 1/5. We need to find the exponent to which 25 must be raised to get 1/5.Express 1/5 as a power: Express 1/5 as a power of 25. Since 25 is 5 squared (5^2), we can express 1/5 as 5^(-1), which is the same as (5^2)^(-1/2).Rewrite using new expression: Rewrite the logarithm using the new expression for 1/5. log_(25)(1/5) becomes log_(25)((5^2)^(-1/2)).Apply power property: Apply the power property of logarithms. The power property of logarithms states that log_b(a^c) = c * log_b(a). Therefore, log_(25)((5^2)^(-1/2)) becomes (-1/2) * log_(25)(5^2).Simplify the logarithm: Simplify the logarithm log_(25)(5^2). Since 25 is 5 squared, log_(25)(5^2) is asking ＂to what power do we raise 25 to get 5 squared?＂ The answer is 1, because 25 to the power of 1 is 25, and we are looking for 5 squared, which is 25.Multiply by the coefficient: Multiply the result from Step 5 by the coefficient from Step 4. (-1/2) * log_(25)(5^2) becomes (-1/2) * 1, which simplifies to -1/2.

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$\log _{25} \frac{1}{5}=$

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Algebra 2

Evaluate logarithms using properties

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\log _{25} \frac{1}{5}=

Recognize base and argument: Recognize the base of the logarithm and the argument. $\newline$ The base of the logarithm is $25$ , and the argument is $\frac{1}{5}$ . We need to find the exponent to which $25$ must be raised to get $\frac{1}{5}$ .
Express $\frac{1}{5}$ as a power: Express $\frac{1}{5}$ as a power of $25$ . $\newline$ Since $25$ is $5$ squared ( $5^2$ ), we can express $\frac{1}{5}$ as $5^{-1}$ , which is the same as $(5^2)^{-\frac{1}{2}}$ .
Rewrite using new expression: Rewrite the logarithm using the new expression for $\frac{1}{5}$ . $\newline$ $\log_{25}\left(\frac{1}{5}\right)$ becomes $\log_{25}\left((5^2)^{-\frac{1}{2}}\right)$ .
Apply power property: Apply the power property of logarithms. $\newline$ The power property of logarithms states that $\log_b(a^c) = c \cdot \log_b(a)$ . Therefore, $\log_{25}((5^2)^{-\frac{1}{2}})$ becomes $(-\frac{1}{2}) \cdot \log_{25}(5^2)$ .
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