Evaluate:log, base, 64, start fraction, 1, divided by, 32, end fractionlog 64​ 321​

1. Break Down the Problem

We need to evaluate the logarithm expression:

$\log_{64} \left(\frac{1}{32}\right)$

This can be rewritten using the properties of logarithms.

2. Relevant Concepts

We know the following logarithmic identities:

• $\log_b (a) = \frac{\log_k (a)}{\log_k (b)}$ for any base $k$.
• The property of logarithms regarding fractions: $\log_b \left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)$.

3. Analysis and Detail

1. Rewrite $\frac{1}{32}$ as $32^{-1}$: $\log_{64}(32^{-1}) = -\log_{64}(32)$

2. We can express $32$ as $2^5$ and $64$ as $2^6$: $-\log_{2^6}(2^5)$

3. Using the change of base formula: $-\frac{\log_2(2^5)}{\log_2(2^6)} = -\frac{5}{6}$

4. Verify and Summarize

Thus, we have: $\log_{64} \left(\frac{1}{32}\right) = -\frac{5}{6}$

Final Answer

$\log_{64} \left(\frac{1}{32}\right) = -\frac{5}{6}$