To simplify the expression [tex]\( 4 \ln x + \ln 3 - \ln x \)[/tex], we can use properties of logarithms and follow these steps:
1. Combine like terms:
Start with the given expression:
[tex]\[
4 \ln x + \ln 3 - \ln x
\][/tex]
2. Combine [tex]\( \ln x \)[/tex] terms:
Notice that [tex]\( 4 \ln x \)[/tex] and [tex]\( - \ln x \)[/tex] are like terms. We can combine them:
[tex]\[
4 \ln x - \ln x = (4 - 1) \ln x = 3 \ln x
\][/tex]
Now the expression becomes:
[tex]\[
3 \ln x + \ln 3
\][/tex]
3. Use the property of logarithms for addition:
Recall the logarithm property that [tex]\( \ln a + \ln b = \ln (a \cdot b) \)[/tex]. Apply this property to combine [tex]\( 3 \ln x \)[/tex] and [tex]\( \ln 3 \)[/tex]:
[tex]\[
3 \ln x + \ln 3 = \ln (x^3) + \ln 3 = \ln (x^3 \cdot 3) = \ln (3 x^3)
\][/tex]
Therefore, the simplified expression is [tex]\( \ln (3 x^3) \)[/tex].
The correct answer is:
[tex]\[
\boxed{\ln (3 x^3)}
\][/tex]
So, the answer is C.
