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

Identify base and argument: Let's identify the base and the argument of the logarithm. The base is 25 and the argument is 1/125. We know that 125 is 5^3, and 25 is 5^2. How can we express 1/125 in terms of the base 25?Rewrite in terms of base: Rewrite 1/125 as (5^3)^(-1) because 1/125 is the reciprocal of 125. Now, we can express this in terms of the base 25. Since 25 is 5^2, we can write the logarithm as log base (5^2) of (5^3)^(-1).Apply power rule: Using the power rule of logarithms, which states that log base b of (a^n) is n * log base b of a, we can simplify the expression. Apply the power rule to the argument (5^3)^(-1) to get -1 * log base (5^2) of (5^3).Simplify further: Now, we can simplify the expression further by recognizing that the base of the logarithm and the base of the argument are powers of the same number. This means that log base (5^2) of (5^3) is simply 3, because (5^2)^3 = 5^6, and the base raised to the power that gives the argument is the result of the logarithm. So, -1 * log base (5^2) of (5^3) becomes -1 * 3.Perform multiplication: Perform the multiplication to find the value of the logarithm. -1 * 3 equals -3. So, log base 25 of 1/125 is -3.

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Evaluate: $\newline$ $\log_{25}\left(\frac{1}{125}\right)$

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

\log_{25}\left(\frac{1}{125}\right)

Identify base and argument: Let's identify the base and the argument of the logarithm. $\newline$ The base is $25$ and the argument is $\frac{1}{125}$ . $\newline$ We know that $125$ is $5^3$ , and $25$ is $5^2$ . $\newline$ How can we express $\frac{1}{125}$ in terms of the base $25$ ?
Rewrite in terms of base: Rewrite $\frac{1}{125}$ as $(5^3)^{-1}$ because $\frac{1}{125}$ is the reciprocal of $125$ . Now, we can express this in terms of the base $25$ . Since $25$ is $5^2$ , we can write the logarithm as $\log_{(5^2)} (5^3)^{-1}$ .
Apply power rule: Using the power rule of logarithms, which states that $\log_b(a^n) = n \cdot \log_b(a)$ , we can simplify the expression. $\newline$ Apply the power rule to the argument $(5^3)^{-1}$ to get $-1 \cdot \log_{5^2}(5^3)$ .
Simplify further: Now, we can simplify the expression further by recognizing that the base of the logarithm and the base of the argument are powers of the same number. $\newline$ This means that $\log_{(5^2)} (5^3)$ is simply $3$ , because $(5^2)^3 = 5^6$ , and the base raised to the power that gives the argument is the result of the logarithm. $\newline$ So, $-1 \times \log_{(5^2)} (5^3)$ becomes $-1 \times 3$ .
Perform multiplication: Perform the multiplication to find the value of the logarithm. $\newline$ $-1 \times 3$ equals $-3$ . $\newline$ So, $\log_{25} \frac{1}{125}$ is $-3$ .