Express Quotient Logarithm: We need to express the logarithm of a quotient in terms of the logarithms of the numerator and the denominator. The logarithm of a quotient rule states that log base b of (a/c) is equal to log base b of a minus log base b of c. Therefore, log base 4 of (1/2) can be written as log base 4 of 1 minus log base 4 of 2.Evaluate Log 1: Now we need to evaluate log base 4 of 1. The logarithm of any number at its own base is 1, so log base 4 of 4 is 1. Since 1 is the multiplicative identity, log base 4 of 1 is 0.Evaluate Log 2: Next, we need to evaluate log base 4 of 2. We know that 4 is 2 squared, so we can express 4 as 2^2. Therefore, log base 4 of 2 is asking us for what power we need to raise 4 to get 2. Since 4 is 2 squared, we need to raise 4 to the power of 1/2 to get 2. So, log base 4 of 2 is 1/2.Combine Results: Now we can combine our results to find the final answer. We have log base 4 of 1/2 equals log base 4 of 1 minus log base 4 of 2, which is 0 - 1/2. Therefore, log base 4 of 1/2 is -1/2.

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$\log_{4}\left(\frac{1}{2}\right)=$

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Precalculus

Quotient property of logarithms

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\log_{4}\left(\frac{1}{2}\right)=

Express Quotient Logarithm: We need to express the logarithm of a quotient in terms of the logarithms of the numerator and the denominator. $\newline$ The logarithm of a quotient rule states that $\log_b\left(\frac{a}{c}\right)$ is equal to $\log_b(a) - \log_b(c)$ . $\newline$ Therefore, $\log_4\left(\frac{1}{2}\right)$ can be written as $\log_4(1) - \log_4(2)$ .
Evaluate Log $1$ : Now we need to evaluate $\log_{4} 1$ . The logarithm of any number at its own base is $1$ , so $\log_{4} 4$ is $1$ . Since $1$ is the multiplicative identity, $\log_{4} 1$ is $0$ .
Evaluate Log $2$ : Next, we need to evaluate $\log_{4} 2$ . We know that $4$ is $2$ squared, so we can express $4$ as $2^2$ . Therefore, $\log_{4} 2$ is asking us for what power we need to raise $4$ to get $2$ . Since $4$ is $2$ squared, we need to raise $4$ to the power of $4$ $1$ to get $2$ . So, $\log_{4} 2$ is $4$ $1$ .
Combine Results: Now we can combine our results to find the final answer. $\newline$ We have $\log_{4} \frac{1}{2}$ equals $\log_{4} 1 - \log_{4} 2$ , which is $0 - \frac{1}{2}$ . $\newline$ Therefore, $\log_{4} \frac{1}{2}$ is $-\frac{1}{2}$ .