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 \(\log_{10} 4\):
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Calculate \(\log_{10} 2.1\): Using a calculator, \(\log_{10} 2.1 \approx 0.322\).
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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:
\[ \boxed{0.535} \]