To solve the problem of finding an equivalent expression for the given expression [tex]\( 4 \ln x + \ln 3 - \ln x \)[/tex], let's proceed step by step:
1. Simplify the given expression:
[tex]\[
4 \ln x + \ln 3 - \ln x
\][/tex]
2. Combine the logarithmic terms involving [tex]\( \ln x \)[/tex]:
[tex]\[
4 \ln x - \ln x = 3 \ln x
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
3 \ln x + \ln 3
\][/tex]
3. Use the properties of logarithms to combine the expression into a single logarithm:
According to the properties of logarithms, specifically the property [tex]\(\ln a + \ln b = \ln (a \cdot b)\)[/tex], we can combine the terms:
[tex]\[
3 \ln x + \ln 3 = \ln (x^3) + \ln 3 = \ln (3x^3)
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\ln (3x^3)
\][/tex]
4. Identify the correct option from the given choices:
A. [tex]\( \ln (3x + 3) \)[/tex]
B. [tex]\( \ln (3x^3) \)[/tex]
C. [tex]\( \ln (x^4 - x + 3) \)[/tex]
D. [tex]\( \ln (11x) \)[/tex]
Comparing the simplified expression [tex]\( \ln (3x^3) \)[/tex] with the options provided, we find that the correct choice is:
B. [tex]\( \ln (3x^3) \)[/tex]
