To simplify [tex]\( 3 \ln 3 - \ln 9 \)[/tex] and express it as a single natural logarithm, follow these steps:
1. Apply the Power Rule of Logarithms:
The power rule states that [tex]\( a \ln b = \ln(b^a) \)[/tex]. Therefore:
[tex]\[
3 \ln 3 = \ln(3^3) = \ln 27
\][/tex]
2. Transform the Expression:
Now our expression [tex]\( 3 \ln 3 - \ln 9 \)[/tex] becomes:
[tex]\[
\ln 27 - \ln 9
\][/tex]
3. Apply the Quotient Rule of Logarithms:
The quotient rule states that [tex]\( \ln a - \ln b = \ln \left(\frac{a}{b}\right)\)[/tex]. Therefore, we can combine the logarithms as follows:
[tex]\[
\ln 27 - \ln 9 = \ln \left(\frac{27}{9}\right)
\][/tex]
4. Simplify the Fraction:
Simplify [tex]\( \frac{27}{9} \)[/tex]:
[tex]\[
\frac{27}{9} = 3
\][/tex]
5. Combine the Results:
Putting it all together:
[tex]\[
\ln \left(\frac{27}{9}\right) = \ln 3
\][/tex]
Therefore, the original expression [tex]\( 3 \ln 3 - \ln 9 \)[/tex] simplifies to [tex]\( \ln 3 \)[/tex].
So, the correct answer is:
[tex]\( \ln 3 \)[/tex]
