Question

Log base of 4 2.1 to the nearest thousandth

Answer 1

To calculate the logarithm of 2.1 with base 4, you can use the change of base formula:

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

Here, we can use base 10 or the natural logarithm (base \(e\)). Using base 10, the formula becomes:

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

Now, let's calculate \( \log_{10}(2.1) \) and \( \log_{10}(4) \):

1. \( \log_{10}(2.1) \approx 0.3222 \)
2. \( \log_{10}(4) = \log_{10}(2^2) = 2 \cdot \log_{10}(2) \approx 2 \cdot 0.3010 = 0.6020 \)

Now we can plug these values into the change of base formula:

\[
\log_4(2.1) \approx \frac{0.3222}{0.6020} \approx 0.535
\]

Thus, rounding to the nearest thousandth, we get:

\[
\log_4(2.1) \approx 0.535
\]