To find the equivalent expression to the given expression [tex]\(4 \ln x + \ln 3 - \ln x\)[/tex], we need to simplify the logarithmic terms step by step.
1. Combine like terms:
We start by focusing on the terms that involve [tex]\(\ln x\)[/tex]. Specifically, we have [tex]\(4 \ln x\)[/tex] and [tex]\(- \ln x\)[/tex].
[tex]\[
4 \ln x - \ln x = (4 - 1) \ln x = 3 \ln x
\][/tex]
2. Re-write the expression:
After combining the logarithmic terms, the expression simplifies to:
[tex]\[
3 \ln x + \ln 3
\][/tex]
3. Combine logarithms:
Now, we use the logarithm property that states: [tex]\(\ln a + \ln b = \ln(ab)\)[/tex]. Applying this rule, we combine [tex]\(3 \ln x\)[/tex] and [tex]\(\ln 3\)[/tex] into a single logarithm:
[tex]\[
3 \ln x + \ln 3 = \ln (x^3) + \ln 3 = \ln (3 \cdot x^3) = \ln (3x^3)
\][/tex]
Therefore, the expression [tex]\(4 \ln x + \ln 3 - \ln x\)[/tex] is equivalent to [tex]\(\ln \left(3 x^3\right)\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{\ln \left(3 x^3\right)}
\][/tex]
So the correct choice is:
[tex]\[
\boxed{\text{D}}
\][/tex]
