Question

What value is equivalent to log42.1
log
4
⁡
2.1
to the nearest thousandth? Use the change of base formula to determine your answer.(1 point)
Responses

0.535
0.535

1.868
1.868

0.081
0.081

0.287

Answer 1

To find the value of \( \log_4{2.1} \) using the change of base formula, you can use the following formula:

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

For this problem, let's use base 10 (common logarithm):

\[
\log_4{2.1} = \frac{\log_{10}{2.1}}{\log_{10}{4}}
\]

Now, we'll calculate each logarithm:

1. \( \log_{10}{2.1} \) is approximately \( 0.322 \).
2. \( \log_{10}{4} = \log_{10}{(2^2)} = 2 \cdot \log_{10}{2} \). Since \( \log_{10}{2} \approx 0.301 \), then \( \log_{10}{4} \approx 2 \cdot 0.301 = 0.602 \).

Now, plug in these values into the change of base formula:

\[
\log_4{2.1} \approx \frac{0.322}{0.602} \approx 0.535
\]

Thus, the value of \( \log_4{2.1} \) to the nearest thousandth is approximately **0.535**.