Question

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

0.081
0.081

0.535
0.535

0.287
0.287

1.868

Answer 1

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

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

where \(k\) can be any positive number different from 1. A common choice is to use base 10 or base \(e\) for logarithms.

Using base 10, we apply the formula like this:

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

Now, we need to calculate \(\log_{10} 2.1\) and \(\log_{10} 4\):

1. **Calculate \(\log_{10} 2.1\):**
Using a calculator, \(\log_{10} 2.1 \approx 0.322\).

2. **Calculate \(\log_{10} 4\):**
Since \(4 = 2^2\), we have:
\(\log_{10} 4 = \log_{10} (2^2) = 2 \cdot \log_{10} 2\).
Using a calculator, \(\log_{10} 2 \approx 0.301\), thus:
\(\log_{10} 4 \approx 2 \cdot 0.301 = 0.602\).

Now, substituting these values into the change of base formula gives:

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

Thus, the value of \(\log_4 2.1\) to the nearest thousandth is:

\[
\boxed{0.535}
\]