Evaluate logarithm of 1/2: We need to evaluate the logarithm of 1/2 with base 4. The logarithm of a number is the exponent to which the base must be raised to produce that number.Express 1/2 as a power of 4: First, let's express 1/2 as a power of 4. Since 4 is 2^2, we can write 1/2 as 2^(-1). This is because 1/2 is the reciprocal of 2, and the reciprocal of a number is equal to that number raised to the power of -1.Rewrite logarithm using power of 4: Now, we can rewrite the logarithm using the power of 4. Since 2 is the square root of 4, we can express 2 as 4^(1/2). Therefore, 2^(-1) can be written as (4^(1/2))^(-1).Simplify using properties of exponents: Using the properties of exponents, (4^(1/2))^(-1) simplifies to 4^(-1/2). This is because when you raise a power to a power, you multiply the exponents.Rewrite logarithm as log base 4: Now we can rewrite the original logarithm as log_(4)(4^(-1/2)).Simplify using property of logarithms: Using the property of logarithms that log_b(b^x) = x, we can simplify log_(4)(4^(-1/2)) to just -1/2.Therefore, the value of log_(4)(1/2) is -1/2.

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$\log _{4} \frac{1}{2}=$

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Algebra 2

Evaluate logarithms using properties

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\log _{4} \frac{1}{2}=

Evaluate logarithm of $\frac{1}{2}$ : We need to evaluate the logarithm of $\frac{1}{2}$ with base $4$ . The logarithm of a number is the exponent to which the base must be raised to produce that number.
Express $\frac{1}{2}$ as a power of $4$ : First, let's express $\frac{1}{2}$ as a power of $4$ . Since $4$ is $2^2$ , we can write $\frac{1}{2}$ as $2^{-1}$ . This is because $\frac{1}{2}$ is the reciprocal of $2$ , and the reciprocal of a number is equal to that number raised to the power of $4$ $0$ .
Rewrite logarithm using power of $4$ : Now, we can rewrite the logarithm using the power of $4$ . Since $2$ is the square root of $4$ , we can express $2$ as $4^{1/2}$ . Therefore, $2^{-1}$ can be written as $(4^{1/2})^{-1}$ .
Simplify using properties of exponents: Using the properties of exponents, $(4^{\frac{1}{2}})^{-1}$ simplifies to $4^{-\frac{1}{2}}$ . This is because when you raise a power to a power, you multiply the exponents.
Rewrite logarithm as log base $4$ : Now we can rewrite the original logarithm as $\log_{4}(4^{-\frac{1}{2}})$ .
Simplify using property of logarithms: Using the property of logarithms that $\log_b(b^x) = x$ , we can simplify $\log_{4}(4^{-\frac{1}{2}})$ to just $-\frac{1}{2}$ .
Simplify using property of logarithms: Using the property of logarithms that $\log_b(b^x) = x$ , we can simplify $\log_{4}(4^{-\frac{1}{2}})$ to just $-\frac{1}{2}$ .Therefore, the value of $\log_{4}(\frac{1}{2})$ is $-\frac{1}{2}$ .