Evaluate the logarithm.

Round your answer to the nearest thousandth.
\[\log_{3}(0.2)\approx\]

1 answer

To evaluate the logarithm \(\log_{3}(0.2)\), we can use the change of base formula. The change of base formula states:

\[ \log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)} \]

for any positive \(k\), which is usually chosen to be 10 or \(e\) (the base of natural logarithms). Here, we will use base 10:

\[ \log_{3}(0.2) = \frac{\log_{10}(0.2)}{\log_{10}(3)} \]

Next, we need to calculate \(\log_{10}(0.2)\) and \(\log_{10}(3)\).

Using a calculator:

  1. \(\log_{10}(0.2) \approx -0.699\)
  2. \(\log_{10}(3) \approx 0.477\)

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

\[ \log_{3}(0.2) \approx \frac{-0.699}{0.477} \]

Calculating the division:

\[ \frac{-0.699}{0.477} \approx -1.465 \]

Rounding this to the nearest thousandth, we get:

\[ \log_{3}(0.2) \approx -1.465 \]

Thus, the final answer is:

\[ \boxed{-1.465} \]