Question

Using the power rule of logarithms, how can log(1.125) be rewritten?
1)-3log(5)
2)log_{5}(1/3)
3)[log(1)]/[log(125)]
4)5log(1/3)

Answer 1

To rewrite \(\log(1.125)\) using the power rule of logarithms, first, let's express \(1.125\) in a more manageable form.

Notice that:
\[
1.125 = \frac{9}{8} = \frac{3^2}{2^3}
\]

Using the properties of logarithms, we can express \(\log(1.125)\) as:
\[
\log(1.125) = \log\left(\frac{3^2}{2^3}\right)
\]

Using the logarithm property for division:
\[
\log\left(\frac{a}{b}\right) = \log(a) - \log(b)
\]
we have:
\[
\log(1.125) = \log(3^2) - \log(2^3) = 2\log(3) - 3\log(2)
\]

Now, we will evaluate the options provided to determine if any of them match this expression.

1) \(-3\log(5)\) – does not match.
2) \(\log_{5}(1/3)\) – does not match.
3) \(\frac{\log(1)}{\log(125)}\) – since \(\log(1) = 0\), this would equal \(0\), which does not match.
4) \(5\log(1/3)\) – does not match either.

Based on this analysis, none of the options provided accurately rewrite \(\log(1.125)\) according to the power rule of logarithms.

To examine if any provided options could perhaps be reinterpreted:

- \(-3\log(5)\) means a negative multiple of a different base.
- \(\log_{5}(1/3)\) would indicate a logarithm concerning different base and argument.
- \(\log(1)\) is \(0\), making choice 3 zero irrelevant.
- \(5\log(1/3)\) is not the correct simplification obtainable from \(\log(1.125)\).

Therefore, we conclude that none of the given choices present a valid transformation of \(\log(1.125)\) using the logarithmic properties as derived.