Use properties of logarithms to expand the logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible.

[tex]\[ \ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right] \][/tex]

[tex]\[ \ln \left[\frac{x^3 \sqrt{x^2+6}}{(x+6)^6}\right] = \qquad \][/tex]



Answer :

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]