To find \( \log_7(10 \cdot 20) \), we can use the Product Rule of Logarithms, which states that:
\[ \log_b(m \cdot n) = \log_b(m) + \log_b(n) \]
In this case, we can express \( \log_7(10 \cdot 20) \) as:
\[ \log_7(10 \cdot 20) = \log_7(10) + \log_7(20) \]
We are given the approximate values of \( \log_7(10) \) and \( \log_7(20) \):
- \( \log_7(10) \approx 1.183 \)
- \( \log_7(20) \approx 1.540 \)
Now, substituting these values into the equation, we have:
\[ \log_7(10 \cdot 20) \approx 1.183 + 1.540 \]
Calculating this sum:
\[ 1.183 + 1.540 = 2.723 \]
Thus, the value of \( \log_7(10 \cdot 20) \) is approximately \( 2.723 \).
Therefore, the correct response is:
2.723