To express [tex]\( 3 \ln 2 + 2 \ln 3 \)[/tex] as a single logarithm, we can utilize properties of logarithms, specifically the power rule and product rule.
1. Power Rule: The power rule states that [tex]\( a \ln(x) = \ln(x^a) \)[/tex]. Let's apply this rule to each term in the expression:
[tex]\[
3 \ln 2 = \ln(2^3)
\][/tex]
[tex]\[
2 \ln 3 = \ln(3^2)
\][/tex]
2. Simplification: Next, we simplify the expressions within the logarithms:
[tex]\[
\ln(2^3) = \ln(8)
\][/tex]
[tex]\[
\ln(3^2) = \ln(9)
\][/tex]
3. Product Rule: The product rule for logarithms states that [tex]\( \ln(a) + \ln(b) = \ln(ab) \)[/tex]. We now combine the two logarithmic expressions:
[tex]\[
\ln(8) + \ln(9) = \ln(8 \times 9)
\][/tex]
4. Calculation: Multiply the numbers inside the logarithm:
[tex]\[
8 \times 9 = 72
\][/tex]
5. Final Expression: Thus, the expression [tex]\( 3 \ln 2 + 2 \ln 3 \)[/tex] as a single logarithm is:
[tex]\[
\ln(72)
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\ln(72)}
\][/tex]
