The logarithm of \( 3^{-1} \) depends on the base of the logarithm. Assuming we're working with the common logarithm (base 10), natural logarithm (base \(e\)), or logarithms with other specific bases, the property of logarithms and exponents can be applied.

For any base \(b\), the logarithm has the property:

\[
\log_b \left( 3^{-1} \right) = \log_b \left( \frac{1}{3} \right) = -\log_b(3)
\]

Therefore:
1. If the base is 10 (common logarithm), then:
   \[
   \log_{10} \left( 3^{-1} \right) = -\log_{10}(3)
   \]
2. If the base is \(e\) (natural logarithm), then:
   \[
   \ln \left( 3^{-1} \right) = -\ln(3)
   \]

Thus, the logarithm of \( 3^{-1} \) is the negative of the logarithm of 3, regardless of the base.

Would you like to see details on specific bases or have more questions?

Here are 5 related questions:
1. How would you calculate \( \log_2(3^{-1}) \)?
2. What is the value of \( \log_{10}(3) \)?
3. How does the change of base formula work for logarithms?
4. What is \( \log_b(3^{-2}) \)?
5. How can logarithms be used in solving exponential equations?

**Tip:** Remember, a logarithm essentially answers the question: "To what power must the base be raised to produce a given number?"

Explore how to find the logarithm of 3^-1. Learn about the logarithmic properties of exponents, including the power rule and how it applies to negative exponents. Suitable for students in Grades 9-12.

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