To find the value of \( \log_2{13} \) using the change of base formula, we use:
\[ \log_b{a} = \frac{\log_k{a}}{\log_k{b}} \]
where \( k \) can be any positive number. A common choice is to use base 10 (common logarithm) or base \( e \) (natural logarithm). Let's use the common logarithm (base 10):
\[ \log_2{13} = \frac{\log_{10}{13}}{\log_{10}{2}} \]
Using a calculator:
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Calculate \( \log_{10}{13} \):
- \( \log_{10}{13} \approx 1.113943 \)
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Calculate \( \log_{10}{2} \):
- \( \log_{10}{2} \approx 0.301030 \)
Now, substitute these values into the formula:
\[ \log_2{13} \approx \frac{1.113943}{0.301030} \approx 3.704 \]
Rounding to the nearest thousandth gives:
\[ \log_2{13} \approx 3.704 \]
Among the options provided:
- \( 0.557 \)
- \( 0.270 \)
- \( 0.813 \)
- \( 3.700 \)
The closest value to \( 3.704 \) is 3.700.
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