What value is equivalent to log42.1

Question

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

0.081
0.081

1.868
1.868

0.535
0.535

0.287

Answer 1

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

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

where \(k\) can be any positive number (commonly we use 10 or \(e\)).

Using base 10:

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

Calculating each logarithm using a calculator:

1. \(\log_{10}(2.1) \approx 0.322\)
2. \(\log_{10}(4) \approx 0.602\)

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

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

Thus, \(\log_4(2.1)\) to the nearest thousandth is approximately 0.535.