Question

Rewrite log(1/81) with a negative exponent, then simplify using the Power Rule of Logarithms.

Answer 1

To rewrite \(\log\left(\frac{1}{81}\right)\) with a negative exponent, we can express \(\frac{1}{81}\) as \(81^{-1}\). Thus, we have:

\[
\log\left(\frac{1}{81}\right) = \log(81^{-1})
\]

Next, we apply the Power Rule of Logarithms, which states that \(\log(a^b) = b \cdot \log(a)\). In this case, \(a\) is \(81\) and \(b\) is \(-1\):

\[
\log(81^{-1}) = -1 \cdot \log(81)
\]

Therefore, we can simplify this expression as:

\[
\log\left(\frac{1}{81}\right) = -\log(81)
\]

This is the simplified form that uses a negative exponent and the Power Rule of Logarithms.