Question

Evaluate the logarithm.
Round your answer to the nearest thousandth.

\[\log_{5}\left(\dfrac{1}{1000}\right)\approx\]

Answer 1

To evaluate \(\log_{5}\left(\dfrac{1}{1000}\right)\), we first express \(\dfrac{1}{1000}\) as a power of 10:

\[
\dfrac{1}{1000} = 10^{-3}.
\]

Now we can rewrite the logarithm:

\[
\log_{5}\left(\dfrac{1}{1000}\right) = \log_{5}(10^{-3}) = -3 \log_{5}(10).
\]

Next, we can use the change of base formula to compute \(\log_{5}(10)\):

\[
\log_{5}(10) = \frac{\log_{10}(10)}{\log_{10}(5)}.
\]

Since \(\log_{10}(10) = 1\), we have:

\[
\log_{5}(10) = \frac{1}{\log_{10}(5)}.
\]

Now, let's calculate \(\log_{10}(5)\). By using the known value of \(\log_{10}(2) \approx 0.3010\) and \(\log_{10}(10) = 1\), we can derive \(\log_{10}(5)\):

\[
\log_{10}(5) = \log_{10}\left(\frac{10}{2}\right) = \log_{10}(10) - \log_{10}(2) \approx 1 - 0.3010 = 0.6990.
\]

Now substituting this back, we find:

\[
\log_{5}(10) \approx \frac{1}{0.6990} \approx 1.4307.
\]

Therefore,

\[
\log_{5}\left(\dfrac{1}{1000}\right) = -3 \log_{5}(10) \approx -3 \times 1.4307 \approx -4.2921.
\]

Finally, rounding to the nearest thousandth, we get:

\[
\log_{5}\left(\dfrac{1}{1000}\right) \approx -4.292.
\]

Thus, the final answer is:

\[
\boxed{-4.292}.
\]