$\log _{8} \frac{1}{32}=$

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Algebra 2

Evaluate logarithms using properties

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\log _{8} \frac{1}{32}=

Recognize logarithm of $1$ : We are asked to find the value of the logarithm of $\frac{1}{32}$ with base $8$ . The first step is to recognize that the logarithm of $1$ to any base is $0$ , because any number to the power of $0$ is $1$ . $\newline$ $\log_{8}(1) = 0$
Express $\frac{1}{32}$ as a power of $8$ : Now we need to consider the $\frac{1}{32}$ part. We can express $32$ as a power of $8$ to use the quotient property of logarithms, which states that the logarithm of a quotient is the difference of the logarithms. $\newline$ $32$ can be written as $2^5$ , and since $8$ is $2^3$ , we can rewrite $32$ as $8$ $0$ .
Simplify the expression: Using the property of exponents $(a^b)^c = a^{(b*c)}$ , we can simplify $(2^3)^{\frac{5}{3}}$ to $2^{(3*\frac{5}{3})}$ , which simplifies to $2^5$ , confirming our previous statement that $32$ is $2^5$ .
Write as difference of logarithms: Now we can write the original expression as the difference of two logarithms: $\newline$ $\log_{8}(1) - \log_{8}(32)$ $\newline$ Since we already know $\log_{8}(1) = 0$ , we only need to evaluate $\log_{8}(32)$ .
Express $32$ as a power of $8$ : We can express $32$ as $8$ to the power of some number. Since $8$ is $2^3$ and $32$ is $2^5$ , we can find the exponent by solving for $x$ in $8^x = 32$ . $\newline$ $8$ $0$
Solve for the exponent: Solving the equation $3^x = 5$ gives us $x = \frac{5}{3}$ . Therefore, $3^2$ is $8$ to the power of $\frac{5}{3}$ , and we can write: $\newline$ $\log_{8}(32) = \log_{8}(8^{\frac{5}{3}})$
Apply power property of logarithms: Using the power property of logarithms, which states that $\log_b(a^c) = c \cdot \log_b(a)$ , we get: $\newline$ $\log_{8}(8^{\frac{5}{3}}) = \frac{5}{3} \cdot \log_{8}(8)$
Evaluate $\log_{8}(8)$ : Since the logarithm of a number to the same base is $1$ , $\log_{8}(8) = 1$ . Therefore: $\newline$ $\frac{5}{3} \cdot \log_{8}(8) = \frac{5}{3} \cdot 1 = \frac{5}{3}$
Combine results to find the value: Now we can combine our results to find the value of the original expression: $\log_{8}(\frac{1}{32}) = \log_{8}(1) - \log_{8}(32) = 0 - \frac{5}{3} = -\frac{5}{3}$