Question

Evaluate the logarithm.
Round your answer to the nearest thousandth.
\[\log_{4}(0.6)\approx\]

Answer 1

To evaluate the logarithm \(\log_{4}(0.6)\), we can use the change of base formula for logarithms, which states:

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

For our case, we can use base \(10\) (common logarithm) or base \(e\) (natural logarithm). Here, we'll use base \(10\):

\[
\log_{4}(0.6) = \frac{\log_{10}(0.6)}{\log_{10}(4)}
\]

Now, we need to calculate \(\log_{10}(0.6)\) and \(\log_{10}(4)\):

1. Calculate \(\log_{10}(0.6)\):
Using a calculator, we find:
\[
\log_{10}(0.6) \approx -0.2218
\]

2. Calculate \(\log_{10}(4)\):
Since \(4 = 2^2\), we can use the property of logarithms:
\[
\log_{10}(4) = 2 \log_{10}(2)
\]
Using a calculator, we find \(\log_{10}(2) \approx 0.3010\), thus:
\[
\log_{10}(4) \approx 2 \times 0.3010 \approx 0.6020
\]

Now we can substitute these values back into our formula:

\[
\log_{4}(0.6) \approx \frac{-0.2218}{0.6020}
\]

Calculating this gives:

\[
\log_{4}(0.6) \approx -0.368
\]

Finally, rounding to the nearest thousandth, we get:

\[
\log_{4}(0.6) \approx -0.368
\]

Thus, the answer is:

\[
\boxed{-0.368}
\]