To calculate \( \log_2{13} \) using the change of base formula, we can use the formula:
\[ \log_b{a} = \frac{\log_k{a}}{\log_k{b}} \]
where \( k \) can be any positive number different from 1. A common choice for \( k \) is 10 or \( e \) (for natural logarithms). Here, we'll use base 10:
\[ \log_2{13} = \frac{\log_{10}{13}}{\log_{10}{2}} \]
Now, we can look up the logarithm values:
- \( \log_{10}{13} \approx 1.113943 \)
- \( \log_{10}{2} \approx 0.30103 \)
Now substitute these values into the formula:
\[ \log_2{13} \approx \frac{1.113943}{0.30103} \approx 3.702 \]
Rounding to the nearest thousandth gives us:
\[ \log_2{13} \approx 3.702 \]
From the provided options, the equivalent value of \( \log_2{13} \) to the nearest thousandth is 3.700.