The expression given is:

$4 \ln(x + 6) - 3 \ln(x)$

To simplify this expression using the properties of logarithms, we can follow these steps:

Step 1: Apply the power rule of logarithms.

The power rule states that $a \ln(b) = \ln(b^a)$. So, we can rewrite both terms:

$4 \ln(x + 6) = \ln((x + 6)^4)$ $3 \ln(x) = \ln(x^3)$

Step 2: Apply the difference rule of logarithms.

The difference of two logarithms can be rewritten as the logarithm of a quotient:

$\ln(A) - \ln(B) = \ln\left(\frac{A}{B}\right)$

Thus, applying this rule:

$\ln((x + 6)^4) - \ln(x^3) = \ln\left(\frac{(x + 6)^4}{x^3}\right)$

Final simplified expression:

$\ln\left(\frac{(x + 6)^4}{x^3}\right)$

This is the condensed logarithmic expression. If you need further details or have any questions, feel free to ask!

Here are five related questions to deepen your understanding:

1. What is the power rule for logarithms, and how is it derived?
2. How does the quotient rule of logarithms apply to more complex expressions?
3. How would the properties of logarithms change if the base were different (e.g., log base 10)?
4. How would you expand logarithmic expressions instead of condensing them?
5. How can logarithmic expressions be useful in solving exponential equations?

Tip: Always check if you can apply multiple logarithmic properties, like the product, quotient, and power rules, to simplify expressions step by step.