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Evaluate:

log_(16)4
Answer:

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

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Evaluate logarithms

Full solution

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 $\frac{1}{2}$ .

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