Sure! Let's use the properties of logarithms to expand the expression [tex]\(\ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right]\)[/tex].
### Step-by-step Solution
1. Original Expression:
[tex]\[
\ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right]
\][/tex]
2. Apply the Quotient Rule:
Using the logarithm property [tex]\(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)[/tex], we can separate the logarithm of the fraction:
[tex]\[
\ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right] = \ln \left[x^3 \sqrt{x^2+6}\right] - \ln \left[(x+6)^6\right]
\][/tex]
3. Apply the Product Rule:
Using the logarithm property [tex]\(\ln(ab) = \ln(a) + \ln(b)\)[/tex], we can separate the logarithm of the product in the numerator:
[tex]\[
\ln \left[x^3 \sqrt{x^2+6}\right] = \ln (x^3) + \ln (\sqrt{x^2+6})
\][/tex]
4. Simplify Further:
Using the power rule [tex]\(\ln(a^b) = b\ln(a)\)[/tex], we can simplify each logarithm:
[tex]\[
\ln(x^3) = 3\ln(x)
\][/tex]
[tex]\[
\ln(\sqrt{x^2+6}) = \ln((x^2+6)^{1/2}) = \frac{1}{2}\ln(x^2+6)
\][/tex]
For the denominator:
[tex]\[
\ln \left[(x+6)^6\right] = 6\ln(x+6)
\][/tex]
5. Combine All the Parts:
Putting it all together, we combine the separated and simplified parts:
[tex]\[
\ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right] = 3\ln(x) + \frac{1}{2}\ln(x^2+6) - 6\ln(x+6)
\][/tex]
So, the expanded form of the given logarithmic expression is:
[tex]\[
\boxed{3\ln(x) + \frac{1}{2}\ln(x^2 + 6) - 6\ln(x + 6)}
\][/tex]
