Identify base and argument: Identify the base and the argument of the logarithm. We are dealing with a logarithm with base 2 and the argument is 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 6th power (2^6), 1/64 can be written as 2^(-6).Rewrite logarithm with new expression: Rewrite the logarithm using the new expression for 1/64. log_(2)(1/64) becomes log_(2)(2^(-6)).Apply power property of logarithms: Apply the power property of logarithms. The power property states that log_b(a^c) = c * log_b(a). Therefore, log_(2)(2^(-6)) equals -6 * log_(2)(2).Evaluate logarithm of 2: Evaluate log_(2)(2). 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. Since log_(2)(2) is 1, -6 * log_(2)(2) is -6 * 1, which equals -6.

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

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Algebra 2

Evaluate logarithms using properties

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\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 $-6$ .