Evaluate: log_(125)25 Answer:

Recognize Relationship: Recognize the relationship between the base of the logarithm and the number. The base of the logarithm is 125, and the number is 25. We know that 125 is 5^3 and 25 is 5^2.Express as Power: Express the number 25 as a power of the base 125. Since 125 is 5^3, we can express 25 as (5^3)^(2/3) because (5^3)^(2/3) = 5^(3*(2/3)) = 5^2 = 25.Apply Change of Base: Apply the change of base formula for logarithms. Using the change of base formula, we can write log_(125)25 as log_(5^3)(5^2).Simplify Using Power Rule: Simplify the logarithm using the power rule. The power rule of logarithms states that log_b(a^c) = c*log_b(a). Therefore, log_(5^3)(5^2) can be simplified to (2/3)*log_(5^3)(5).Recognize Same Base: Recognize that the base and the number inside the logarithm are now the same. Since the base is 5^3 and the number is 5, we can simplify log_(5^3)(5) to 1/3 because 5 is the cube root of 5^3.Multiply Final Result: Multiply the result from Step 5 by the coefficient from Step 4. Multiplying (1/3) by (2/3) gives us the final result of the logarithm. (2/3)*(1/3) = 2/9.

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

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Precalculus

Product property of logarithms

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Q. Evaluate:

\newline

\log _{125} 25

\newline

Answer:

Recognize Relationship: Recognize the relationship between the base of the logarithm and the number. $\newline$ The base of the logarithm is $125$ , and the number is $25$ . We know that $125$ is $5^3$ and $25$ is $5^2$ .
Express as Power: Express the number $25$ as a power of the base $125$ . $\newline$ Since $125$ is $5^3$ , we can express $25$ as $(5^3)^{\frac{2}{3}}$ because $(5^3)^{\frac{2}{3}} = 5^{3*\frac{2}{3}} = 5^2 = 25$ .
Apply Change of Base: Apply the change of base formula for logarithms. $\newline$ Using the change of base formula, we can write $\log_{125}25$ as $\log_{5^3}(5^2)$ .
Simplify Using Power Rule: Simplify the logarithm using the power rule. $\newline$ The power rule of logarithms states that $\log_b(a^c) = c\log_b(a)$ . Therefore, $\log_{5^3}(5^2)$ can be simplified to $(\frac{2}{3})\log_{5^3}(5)$ .
Recognize Same Base: Recognize that the base and the number inside the logarithm are now the same. Since the base is $5^3$ and the number is $5$ , we can simplify $\log_{5^3}(5)$ to $\frac{1}{3}$ because $5$ is the cube root of $5^3$ .
Multiply Final Result: Multiply the result from Step $5$ by the coefficient from Step $4$ . $\newline$ Multiplying $(\frac{1}{3})$ by $(\frac{2}{3})$ gives us the final result of the logarithm. $\newline$ $(\frac{2}{3})\ast(\frac{1}{3}) = \frac{2}{9}$ .