Question

simplify. enter the result as a single logarithm with a coefficient of 1.
to enter the log, click in the answer cell, then click on the small yellow arrow on the right.
then click the functions tab to access the log. if needed, you can change the base.
using the keyboard, you can also enter \\( \log _ { 3 } \left( \frac { x ^ { 2 } } { 2 } \
ight) \\) as log_3(x^2/2).
\\( \log _ { 3 } \left( 6 x ^ { 9 } \
ight) - \log _ { 3 } \left( x ^ { 6 } \
ight) = \\)
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Explanation:

Step1: Apply the quotient rule of logarithms

The quotient rule states that \(\log_a M-\log_a N = \log_a\frac{M}{N}\).
For \(\log_3(6x^{9})-\log_3(x^{6})\), we have \(M = 6x^{9}\) and \(N=x^{6}\). So, \(\log_3(6x^{9})-\log_3(x^{6})=\log_3\frac{6x^{9}}{x^{6}}\).

Step2: Simplify the fraction

Using the rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), for \(\frac{6x^{9}}{x^{6}}\), we get \(6x^{9-6}=6x^{3}\).

Answer:

\(\log_3(6x^{3})\)