Question
Rewrite log (1/81) with a negative exponent then simplify using the power rule of logarithms
Answers
GPT-4o mini
To rewrite \(\log\left(\frac{1}{81}\right)\) with a negative exponent, we can express \(\frac{1}{81}\) as \(81^{-1}\). Therefore, we have:
\[
\log\left(\frac{1}{81}\right) = \log\left(81^{-1}\right)
\]
Now, we can apply the power rule of logarithms, which states that \(\log(a^b) = b \cdot \log(a)\). This gives us:
\[
\log\left(81^{-1}\right) = -1 \cdot \log(81) = -\log(81)
\]
Thus, the rewritten and simplified expression for \(\log\left(\frac{1}{81}\right)\) is:
\[
-\log(81)
\]
\[
\log\left(\frac{1}{81}\right) = \log\left(81^{-1}\right)
\]
Now, we can apply the power rule of logarithms, which states that \(\log(a^b) = b \cdot \log(a)\). This gives us:
\[
\log\left(81^{-1}\right) = -1 \cdot \log(81) = -\log(81)
\]
Thus, the rewritten and simplified expression for \(\log\left(\frac{1}{81}\right)\) is:
\[
-\log(81)
\]