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

Understand Problem: Understand the problem and identify the properties of logarithms to use. We need to evaluate the logarithm of 1/125 with base 25. We can use the change of base formula for logarithms, which is log base b of a equals log a divided by log b, where all the logarithms are in the same base.Apply Change of Base: Apply the change of base formula. Using the change of base formula, we can write the expression as: log_(25)((1)/(125)) = log((1)/(125)) / log(25) We know that log(25) is log base 10 of 25, and log((1)/(125)) is log base 10 of 1/125.Evaluate Logarithms: Evaluate the logarithms. Since 25 is 5 squared and 125 is 5 cubed, we can rewrite the logarithms as: log((1)/(125)) = log(1) - log(125) = 0 - log(5^3) = 0 - 3log(5) log(25) = log(5^2) = 2log(5)Simplify Expression: Simplify the expression. Now we can divide the two expressions: (log((1)/(125)) / log(25)) = (0 - 3log(5)) / (2log(5)) This simplifies to: (-3log(5)) / (2log(5)) = -3 / 2Check for Errors: Check for any mathematical errors. We have used the properties of logarithms correctly and simplified the expression without any mistakes.

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

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

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

\newline

Answer:

Understand Problem: Understand the problem and identify the properties of logarithms to use. $\newline$ We need to evaluate the logarithm of $\frac{1}{125}$ with base $25$ . We can use the change of base formula for logarithms, which is $\log_{b}a = \frac{\log a}{\log b}$ , where all the logarithms are in the same base.
Apply Change of Base: Apply the change of base formula. $\newline$ Using the change of base formula, we can write the expression as: $\newline$ $\log_{25}\left(\frac{1}{125}\right) = \frac{\log\left(\frac{1}{125}\right)}{\log(25)}$ $\newline$ We know that $\log(25)$ is log base $10$ of $25$ , and $\log\left(\frac{1}{125}\right)$ is log base $10$ of $1$ / $125$ .
Evaluate Logarithms: Evaluate the logarithms. $\newline$ Since $25$ is $5$ squared and $125$ is $5$ cubed, we can rewrite the logarithms as: $\newline$ $\log\left(\frac{1}{125}\right) = \log(1) - \log(125) = 0 - \log(5^3) = 0 - 3\log(5)$ $\newline$ $\log(25) = \log(5^2) = 2\log(5)$
Simplify Expression: Simplify the expression. $\newline$ Now we can divide the two expressions: $\newline$ $\frac{\log\left(\frac{1}{125}\right)}{\log(25)}$ = $\frac{0 - 3\log(5)}{2\log(5)}$ $\newline$ This simplifies to: $\newline$ $\frac{-3\log(5)}{2\log(5)}$ = $-\frac{3}{2}$
Check for Errors: Check for any mathematical errors. We have used the properties of logarithms correctly and simplified the expression without any mistakes.