Question
10) Using the Power Rule of Logarithms, how can log(18) be rewritten?(1 point) Responses 3log−2 3 log minus 2 log−6 log minus 6 −3log2 negative 3 log 2 2log−3
Answers
GPT-4o mini
To rewrite \( \log(18) \) using the Power Rule of Logarithms, we can start by breaking down \( 18 \) into its prime factors:
\[
18 = 2 \times 3^2
\]
Then, using the property of logarithms that states \( \log(a \times b) = \log(a) + \log(b) \), we have:
\[
\log(18) = \log(2 \times 3^2) = \log(2) + \log(3^2)
\]
Now, applying the Power Rule of Logarithms, which states \( \log(a^b) = b \log(a) \), we can rewrite \( \log(3^2) \):
\[
\log(3^2) = 2 \log(3)
\]
Thus, we can combine our results:
\[
\log(18) = \log(2) + 2 \log(3)
\]
However, the responses you provided do not directly represent this result. Amongst those options, if you intended to represent \( 2 \log(3) \) and \( \log(2) \), it seems none of the options directly match the breakdown provided.
If we assume there's a need to express in a different manner or there's a typographical or input error in the options, it would be prudent to evaluate them again or clarify if a specific outcome is required. If you can provide clarity on the choices or the context for the transformation, I can assist better.
\[
18 = 2 \times 3^2
\]
Then, using the property of logarithms that states \( \log(a \times b) = \log(a) + \log(b) \), we have:
\[
\log(18) = \log(2 \times 3^2) = \log(2) + \log(3^2)
\]
Now, applying the Power Rule of Logarithms, which states \( \log(a^b) = b \log(a) \), we can rewrite \( \log(3^2) \):
\[
\log(3^2) = 2 \log(3)
\]
Thus, we can combine our results:
\[
\log(18) = \log(2) + 2 \log(3)
\]
However, the responses you provided do not directly represent this result. Amongst those options, if you intended to represent \( 2 \log(3) \) and \( \log(2) \), it seems none of the options directly match the breakdown provided.
If we assume there's a need to express in a different manner or there's a typographical or input error in the options, it would be prudent to evaluate them again or clarify if a specific outcome is required. If you can provide clarity on the choices or the context for the transformation, I can assist better.