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!