Question

Rewrite log base 4 (9m^3/n) using the quotient rule of logarithms

Answer 1

To rewrite \(\log_4\left(\frac{9m^3}{n}\right)\) using the quotient rule of logarithms, we can apply the following property:

\[
\log_b\left(\frac{A}{B}\right) = \log_b(A) - \log_b(B)
\]

Using this property, we can rewrite the logarithm as follows:

\[
\log_4\left(\frac{9m^3}{n}\right) = \log_4(9m^3) - \log_4(n)
\]

Next, we can further apply the product rule of logarithms to \(\log_4(9m^3)\):

\[
\log_b(AB) = \log_b(A) + \log_b(B)
\]

Therefore, we can break it down as follows:

\[
\log_4(9m^3) = \log_4(9) + \log_4(m^3)
\]

And applying the power rule of logarithms, we rewrite \(\log_4(m^3)\):

\[
\log_4(m^3) = 3\log_4(m)
\]

Putting it all together, we have:

\[
\log_4\left(\frac{9m^3}{n}\right) = \log_4(9) + 3\log_4(m) - \log_4(n)
\]

So the final expression is:

\[
\log_4\left(\frac{9m^3}{n}\right) = \log_4(9) + 3\log_4(m) - \log_4(n)
\]