Evaluate: log_(16)4 Answer:

Identify Base: In log_(16)4, 16 is the base. Rewrite 4 as a power of 16. Since 4 is not a power of 16, we need to find a common base for both 4 and 16. 4 can be written as 2^2 and 16 can be written as 2^4.Rewrite as Power: Rewrite the expression using the common base. log_(16)4 becomes log_(2^4)(2^2).Common Base: Apply the logarithm power rule, which states that log_b(a^c) = c * log_b(a). log_(2^4)(2^2) becomes 2 * log_(2^4)(2).Rewrite Using Base: Evaluate log_(2^4)(2). When the base of the logarithm is the same as the base of the argument, the logarithm equals 1. log_(2^4)(2) is log_(2^4)(2^1), which simplifies to 1/4 because the exponent of the argument (1) is divided by the exponent of the base (4).Apply Power Rule: Multiply the result from Step 4 by the coefficient from Step 3. 2 * 1/4 equals 1/2.

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Evaluate: $\newline$ $\log _{16} 4$ $\newline$ Answer:

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Q. Evaluate:

\newline

\log _{16} 4

\newline

Answer:

Identify Base: In $\log_{16}4$ , $16$ is the base. $\newline$ Rewrite $4$ as a power of $16$ . $\newline$ Since $4$ is not a power of $16$ , we need to find a common base for both $4$ and $16$ . $\newline$ $4$ can be written as $2^2$ and $16$ can be written as $16$ $1$ .
Rewrite as Power: Rewrite the expression using the common base. $\log_{16}4$ becomes $\log_{2^4}(2^2)$ .
Common Base: Apply the logarithm power rule, which states that $\log_b(a^c) = c \cdot \log_b(a)$ . $\log_{2^4}(2^2)$ becomes $2 \cdot \log_{2^4}(2)$ .
Rewrite Using Base: Evaluate $\log_{2^4}(2)$ . $\newline$ When the base of the logarithm is the same as the base of the argument, the logarithm equals $1$ . $\newline$ $\log_{2^4}(2)$ is $\log_{2^4}(2^1)$ , which simplifies to $\frac{1}{4}$ because the exponent of the argument ( $1$ ) is divided by the exponent of the base ( $4$ ).
Apply Power Rule: Multiply the result from Step $4$ by the coefficient from Step $3$ . $\newline$ $2 \times \frac{1}{4}$ equals $\frac{1}{2}$ .