To calculate \( \log_2{13} \) using the change of base formula, we use:
\[ \log_b{a} = \frac{\log_k{a}}{\log_k{b}} \]
We can use base 10 (common logarithm) or base \( e \) (natural logarithm). Let's use base 10:
\[ \log_2{13} = \frac{\log_{10}{13}}{\log_{10}{2}} \]
Now we need to find \( \log_{10}{13} \) and \( \log_{10}{2} \) using a calculator:
- \( \log_{10}{13} \approx 1.113943 \)
- \( \log_{10}{2} \approx 0.301030 \)
Now we plug these values into the formula:
\[ \log_2{13} \approx \frac{1.113943}{0.301030} \approx 3.703 \]
Rounding \( 3.703 \) to the nearest thousandth gives us:
\[ \log_2{13} \approx 3.703 \]
Thus, the correct response is:
3.700