**Evaluate the logarithm.**
*Round your answer to the nearest thousandth.*

\[6\log_{7}(681)\approx\] 


To evaluate the expression \( 6 \log_{7}(681) \) and round the answer to the nearest thousandth, follow these steps:

1. **Change of Base Formula**: 
   Use the change of base formula to convert the logarithm with base 7 to a common logarithm (base 10) or natural logarithm (base \(e\)):
   \[
   \log_{7}(681) = \frac{\log(681)}{\log(7)}
   \]
   
2. **Calculate Individual Logarithms**:
   - \( \log(681) \) using base 10
   - \( \log(7) \) using base 10

3. **Divide the Results**:
   \[
   \log_{7}(681) = \frac{\log(681)}{\log(7)}
   \]

4. **Multiply by 6**:
   \[
   6 \times \log_{7}(681)
   \]

Let's compute the value:

\[
\log(681) \approx 2.833
\]
\[
\log(7) \approx 0.845
\]
\[
\log_{7}(681) \approx \frac{2.833}{0.845} \approx 3.352
\]
\[
6 \times 3.352 \approx 20.112
\]

So, the final rounded answer is:
\[
6 \log_{7}(681) \approx 20.112
\]

Would you like more details, or do you have any questions? Here are some related topics:

1. How to derive the change of base formula.
2. Why logarithms to base 10 and base \(e\) are commonly used.
3. Applications of logarithms in real-world scenarios.
4. The properties of logarithms and their proofs.
5. Differences between natural and common logarithms.

**Tip:** Understanding the properties of logarithms can simplify complex expressions, making it easier to solve them!

Learn how to evaluate the expression 6 times the logarithm base 7 of 681 with a step-by-step solution. This problem involves using the change of base formula and multiplication of logarithms. Suitable for advanced high school students.

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