Evaluate: log_(8)4 Answer:

Recognize Relationship: Recognize the relationship between the base of the logarithm and the number. We have log base 8 of 4, which can be written as log₈(4). We need to find the power to which 8 must be raised to get 4. Since 4 is a power of 2, and 8 is also a power of 2 (2^3), we can express 4 as 2^2. This will help us to simplify the expression using the change of base formula or properties of logarithms.Express as Power: Express 4 as a power of 2 and use the properties of logarithms. We know that 4 = 2^2, so we can write log₈(4) as log₈(2^2). Using the power property of logarithms, which states that log_b(a^c) = c * log_b(a), we can simplify this to 2 * log₈(2).Evaluate log₈(2): Evaluate log₈(2) using the fact that 8 is a power of 2. Since 8 = 2^3, we can use the inverse property of logarithms, which states that log_b(b^c) = c, to find that log₈(2) is the power to which 8 must be raised to get 2. We can see that 8^(1/3) = 2, so log₈(2) = 1/3.Multiply Result: Multiply the result from Step 3 by 2. Now we multiply 2 by 1/3 to get the final answer. 2 * (1/3) = 2/3.

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Evaluate: $\newline$ $\log _{8} 4$ $\newline$ Answer:

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Precalculus

Product property of logarithms

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Q. Evaluate:

\newline

\log _{8} 4

\newline

Answer:

Recognize Relationship: Recognize the relationship between the base of the logarithm and the number. $\newline$ We have $\log_8(4)$ , which can be written as $\log_8(4)$ . We need to find the power to which $8$ must be raised to get $4$ . Since $4$ is a power of $2$ , and $8$ is also a power of $2$ ( $2^3$ ), we can express $4$ as $\log_8(4)$ $0$ . This will help us to simplify the expression using the change of base formula or properties of logarithms.
Express as Power: Express $4$ as a power of $2$ and use the properties of logarithms. $\newline$ We know that $4 = 2^2$ , so we can write $\log_8(4)$ as $\log_8(2^2)$ . Using the power property of logarithms, which states that $\log_b(a^c) = c \cdot \log_b(a)$ , we can simplify this to $2 \cdot \log_8(2)$ .
Evaluate $\log_8(2)$ : Evaluate $\log_8(2)$ using the fact that $8$ is a power of $2$ . $\newline$ Since $8 = 2^3$ , we can use the inverse property of logarithms, which states that $\log_b(b^c) = c$ , to find that $\log_8(2)$ is the power to which $8$ must be raised to get $2$ . We can see that $8^{1/3} = 2$ , so $\log_8(2) = 1/3$ .
Multiply Result: Multiply the result from Step $3$ by $2$ . Now we multiply $2$ by $1/3$ to get the final answer. $2 \times (1/3) = 2/3$ .