What value is equivalent to log213

Question

log
2
⁡
13
to the nearest thousandth? Use the change of base formula to determine your answer.(1 point)
Responses

0.557
0.557

0.813
0.813

3.700
3.700

0.270

Answer 1

To find the value of \( \log_2 13 \) using the change of base formula, you can use the following formula:

\[ \log_b a = \frac{\log_k a}{\log_k b} \]

where \( k \) is any positive number (common choices are 10 or \( e \)). In this case, we can use base 10 (common logarithm):

\[ \log_2 13 = \frac{\log_{10} 13}{\log_{10} 2} \]

Now, using a calculator:

1. Calculate \( \log_{10} 13 \): \[ \log_{10} 13 \approx 1.113943 \]

2. Calculate \( \log_{10} 2 \): \[ \log_{10} 2 \approx 0.301030 \]

Now, substitute these values into the change of base formula:

\[ \log_2 13 \approx \frac{1.113943}{0.301030} \approx 3.703 \]

Rounding this to the nearest thousandth gives \( 3.703 \).

However, this value is not among the provided options (0.557, 0.813, 3.700, 0.270), but \( 3.700 \) is very close and can be considered as the correct answer among the listed choices.

Thus the answer is:

3.700