To simplify the given logarithmic expression:
[tex]\[
\frac{1}{2}[6 \ln (x-3) + \ln x - 2 \ln x]
\][/tex]
we follow these steps:
1. Combine like terms inside the brackets:
Begin by combining the logarithmic terms in the brackets. Given expression inside the brackets is:
[tex]\[
6 \ln (x-3) + \ln x - 2 \ln x
\][/tex]
Simplify [tex]\(\ln x - 2 \ln x\)[/tex] to get:
[tex]\[
\ln x - 2 \ln x = -\ln x
\][/tex]
So, the expression inside the brackets becomes:
[tex]\[
6 \ln (x-3) - \ln x
\][/tex]
2. Distribute the [tex]\(\frac{1}{2}\)[/tex] coefficient:
Next, distribute the [tex]\(\frac{1}{2}\)[/tex] coefficient to each term inside the brackets:
[tex]\[
\frac{1}{2}[6 \ln (x-3) - \ln x]
\][/tex]
This results in:
[tex]\[
\frac{1}{2} \cdot 6 \ln (x-3) - \frac{1}{2} \ln x
\][/tex]
Which simplifies to:
[tex]\[
3 \ln (x-3) - \frac{1}{2} \ln x
\][/tex]
3. Write as a single logarithm:
Apply the property of logarithms that states [tex]\( a \ln b = \ln b^a \)[/tex] to each term:
[tex]\[
3 \ln (x-3) = \ln (x-3)^3 \quad \text{and} \quad \frac{-1}{2} \ln x = \ln x^{-\frac{1}{2}}
\][/tex]
So, the expression becomes:
[tex]\[
\ln (x-3)^3 - \ln x^{\frac{1}{2}}
\][/tex]
4. Combine the logarithms using the subtraction rule:
Recall the property of logarithms that states [tex]\( \ln A - \ln B = \ln \left( \frac{A}{B} \right) \)[/tex]:
[tex]\[
\ln \left( \frac{(x-3)^3}{x^{\frac{1}{2}}} \right)
\][/tex]
Finally, this expression simplifies to:
[tex]\[
\ln \left(\frac{(x-3)^3}{\sqrt{x}}\right)
\][/tex]
Therefore, the correct answer is:
d. [tex]\(\ln \left(\frac{(x-3)^3}{\sqrt{x}}\right)\)[/tex]
