Evaluate: log_(125)((1)/(25)) Answer:

Recognize base and argument: Recognize the base of the logarithm and the argument. We have log base 125 of 1/25. We can use the change of base formula to rewrite this logarithm in terms of common logarithms or natural logarithms. Change of base formula: log_b(a) = log_c(a) / log_c(b), where c is any positive number different from 1.Apply change of base formula: Apply the change of base formula using base 5, which is a common factor of both 125 and 25. log_(125)((1)/(25)) = log_5((1)/(25)) / log_5(125)Evaluate with base 5: Evaluate both logarithms with base 5. Since 125 is 5^3 and 25 is 5^2, we can rewrite the logarithms as: log_5((1)/(25)) = log_5(5^(-2)) and log_5(125) = log_5(5^3)Use power rule to simplify: Use the power rule of logarithms to simplify. The power rule states that log_b(a^n) = n * log_b(a). Therefore, log_5(5^(-2)) = -2 * log_5(5) and log_5(5^3) = 3 * log_5(5)Further simplify: Since log_5(5) is equal to 1, we can further simplify. -2 * log_5(5) = -2 * 1 = -2 and 3 * log_5(5) = 3 * 1 = 3Divide for final answer: Divide the results to get the final answer. log_(125)((1)/(25)) = (-2) / 3

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Evaluate: $\newline$ $\log _{125} \frac{1}{25}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log _{125} \frac{1}{25}

\newline

Answer:

Recognize base and argument: Recognize the base of the logarithm and the argument. $\newline$ We have $\log_{125} \frac{1}{25}$ . We can use the change of base formula to rewrite this logarithm in terms of common logarithms or natural logarithms. $\newline$ Change of base formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is any positive number different from $1$ .
Apply change of base formula: Apply the change of base formula using base $5$ , which is a common factor of both $125$ and $25$ . $\newline$ $\log_{125}\left(\frac{1}{25}\right) = \frac{\log_{5}\left(\frac{1}{25}\right)}{\log_{5}(125)}$
Evaluate with base $5$ : Evaluate both logarithms with base $5$ . $\newline$ Since $125$ is $5^3$ and $25$ is $5^2$ , we can rewrite the logarithms as: $\newline$ $\log_5\left(\frac{1}{25}\right) = \log_5\left(5^{-2}\right)$ and $\log_5(125) = \log_5\left(5^3\right)$
Use power rule to simplify: Use the power rule of logarithms to simplify. $\newline$ The power rule states that $\log_b(a^n) = n \cdot \log_b(a)$ . $\newline$ Therefore, $\log_5(5^{-2}) = -2 \cdot \log_5(5)$ and $\log_5(5^3) = 3 \cdot \log_5(5)$
Further simplify: Since $\log_5(5)$ is equal to $1$ , we can further simplify. $\newline$ $-2 \times \log_5(5) = -2 \times 1 = -2$ and $3 \times \log_5(5) = 3 \times 1 = 3$
Divide for final answer: Divide the results to get the final answer. $\log_{125}\left(\frac{1}{25}\right) = \frac{-2}{3}$