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log_(2)(1)/(64)=

$\log _{2} \frac{1}{64}=$

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Evaluate logarithms using properties

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Q.

\log _{2} \frac{1}{64}=

Identify base and argument: Identify the base and the argument of the logarithm. $\newline$ We are dealing with a logarithm with base $2$ and the argument is $\frac{1}{64}$ .
Express argument as power of $2$ : Recognize that $1/64$ can be expressed as a power of $2$ . Since $64$ is $2$ raised to the $6$ th power ( $2^6$ ), $1/64$ can be written as $2^{-6}$ .
Rewrite logarithm with new expression: Rewrite the logarithm using the new expression for $\frac{1}{64}$ . $\newline$ $\log_{2}\left(\frac{1}{64}\right)$ becomes $\log_{2}(2^{-6})$ .
Apply power property of logarithms: Apply the power property of logarithms. $\newline$ The power property states that $\log_b(a^c) = c \cdot \log_b(a)$ . Therefore, $\log_{2}(2^{-6})$ equals $-6 \cdot \log_{2}(2)$ .
Evaluate logarithm of $2$ : Evaluate $\log_{2}(2)$ . $\newline$ The logarithm of a number to the same base is $1$ . So, $\log_{2}(2)$ is $1$ .
Multiply result by $-6$ : Multiply the result from Step $5$ by $-6$ . $\newline$ Since $\log_{2}(2)$ is $1$ , $-6 \times \log_{2}(2)$ is $-6 \times 1$ , which equals $-6$ .

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