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log_(64)(1)/(4)=

$\log _{64} \frac{1}{4}=$

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Evaluate logarithms using properties

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Q.

\log _{64} \frac{1}{4}=

Identify base and argument: Identify the base of the logarithm and the argument. $\newline$ The base of the logarithm is $64$ , and the argument is $\frac{1}{4}$ .
Express argument as power of base: Express the argument $\frac{1}{4}$ as a power of the base $64$ . $\newline$ Since $64$ is $2$ raised to the $6$ th power ( $64 = 2^6$ ), we need to express $\frac{1}{4}$ in terms of a power of $2$ . We know that $\frac{1}{4}$ is $2$ raised to the power of $64$ $0$ ( $64$ $1$ ).
Express power of base as power of $64$ : Express $2^{-2}$ as a power of $64$ . $\newline$ Since $64$ is $2^6$ , we can write $2^{-2}$ as $(2^6)^{-\frac{1}{3}}$ because $(2^6)^{-\frac{1}{3}} = 2^{6(-\frac{1}{3})} = 2^{-2}$ .
Write logarithm with new expression: Write the logarithm with the new expression for $\frac{1}{4}$ . $\newline$ $\log_{64}\left((2^6)^{-\frac{1}{3}}\right)$
Apply power property of logarithms: Apply the power property of logarithms. $\newline$ The power property of logarithms states that $\log_b(a^c) = c \cdot \log_b(a)$ . Therefore, we have: $\newline$ $\log_{64}((2^6)^{-\frac{1}{3}}) = -\frac{1}{3} \cdot \log_{64}(2^6)$
Evaluate logarithm: Evaluate $\log_{64}(2^6)$ . $\newline$ Since the base of the logarithm $(64)$ is equal to $2^6$ , $\log_{64}(2^6) = 1$ .
Multiply result by exponent: Multiply the result from Step $6$ by the exponent from Step $5$ . $\newline$ $(-\frac{1}{3}) \times 1 = -\frac{1}{3}$
Conclude final answer: Conclude the final answer. $\newline$ The value of $\log_{64}(\frac{1}{4})$ is $-\frac{1}{3}$ .

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