Evaluate: log_(32)((1)/(8)) Answer:

Understand and Apply Change of Base Formula: Understand the problem and apply the change of base formula. We need to evaluate the logarithm of 1/8 with base 32. The change of base formula states that log_base(b)(a) = log_c(a) / log_c(b), where c is a new base we choose. A common choice for the new base is 10 or e, but in this case, we can choose 2 to simplify the calculation since 32 and 8 are both powers of 2.Apply Change of Base with Base 2: Apply the change of base formula using base 2. Using base 2, we can rewrite the original logarithm as log_2(1/8) / log_2(32). We know that 32 is 2^5 and 8 is 2^3.Evaluate Logarithms with Base 2: Evaluate the logarithms with base 2. Since log_2(2^5) = 5 and log_2(2^3) = 3, we can substitute these values into our equation from Step 2. So, we have log_2(1/8) / log_2(32) = 3 / 5.Simplify the Fraction: Simplify the fraction. The fraction 3/5 is already in its simplest form, so this is our final answer.

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Evaluate: $\newline$ $\log _{32} \frac{1}{8}$ $\newline$ Answer:

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Precalculus

Quotient property of logarithms

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Q. Evaluate:

\newline

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

\newline

Answer:

Understand and Apply Change of Base Formula: Understand the problem and apply the change of base formula. $\newline$ We need to evaluate the logarithm of $\frac{1}{8}$ with base $32$ . The change of base formula states that $\log_{\text{base}}(b)(a) = \frac{\log_{c}(a)}{\log_{c}(b)}$ , where $c$ is a new base we choose. A common choice for the new base is $10$ or $e$ , but in this case, we can choose $2$ to simplify the calculation since $32$ and $8$ are both powers of $2$ .
Apply Change of Base with Base $2$ : Apply the change of base formula using base $2$ . Using base $2$ , we can rewrite the original logarithm as $\log_2(\frac{1}{8}) / \log_2(32)$ . We know that $32$ is $2^5$ and $8$ is $2^3$ .
Evaluate Logarithms with Base $2$ : Evaluate the logarithms with base $2$ . $\newline$ Since $\log_2(2^5) = 5$ and $\log_2(2^3) = 3$ , we can substitute these values into our equation from Step $2$ . So, we have $\log_2(1/8) / \log_2(32) = 3 / 5$ .
Simplify the Fraction: Simplify the fraction. $\newline$ The fraction $\frac{3}{5}$ is already in its simplest form, so this is our final answer.