Question
39. Which of the following is the logarithmic form of the equation \( 4^{-3}=\frac{1}{64} \) ? \( \left.\log _{4}\left(\frac{1}{64}\right)=-3\right) \quad \log _{3}(4)=\frac{1}{64} \) \( \log _{\frac{1}{4}}(4)=-3 \)
Ask by Sullivan Carter. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The logarithmic form of the equation \(4^{-3} = \frac{1}{64}\) is \(\log_4\left(\frac{1}{64}\right) = -3\).
Solution
To convert the exponential equation \(4^{-3} = \frac{1}{64}\) into its logarithmic form, we use the definition of a logarithm:
\[
a^b = c \quad \text{is equivalent to} \quad \log_a(c) = b
\]
Applying this to the given equation:
\[
4^{-3} = \frac{1}{64} \quad \Rightarrow \quad \log_4\left(\frac{1}{64}\right) = -3
\]
Let's evaluate the provided options:
1. **\(\log_4\left(\frac{1}{64}\right) = -3\)**
- **Correct.** This directly matches the conversion using the logarithm definition.
2. **\(\log_3(4) = \frac{1}{64}\)**
- **Incorrect.** The base here is 3, which is unrelated to the original equation.
3. **\(\log_{\frac{1}{4}}(4) = -3\)**
- **Incorrect.** To verify:
\[
\left(\frac{1}{4}\right)^{-3} = 4^3 = 64 \neq 4
\]
This does not satisfy the original equation.
**Therefore, the correct logarithmic form of the equation \(4^{-3} = \frac{1}{64}\) is:**
\[
\boxed{\log_4\left(\frac{1}{64}\right) = -3}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
The equation \( 4^{-3}=\frac{1}{64} \) can be rewritten in logarithmic form as \( \log_{4}\left(\frac{1}{64}\right)=-3 \). This expression indicates that 4 raised to the power of -3 equals \( \frac{1}{64} \), illustrating a fundamental property of logarithms. Thinking about logarithmic conversions can be tricky, but here’s a tip: always remember that if you have \( a^b = c \), the logarithmic form will be \( \log_{a}(c) = b \). A common mistake is confusing the base or getting the signs mixed up, so double-check each part to make sure everything aligns correctly!
