We are asked to condense the following logarithmic expression:

$4 \ln(x + 8) - 8 \ln(x)$

Step 1: Apply the power rule

The power rule for logarithms states that:

$a \ln(b) = \ln(b^a)$

So, we apply this rule to both terms:

$4 \ln(x + 8) = \ln((x + 8)^4)$ $-8 \ln(x) = \ln(x^{-8})$

Step 2: Combine the logarithms

Now we have:

$\ln((x + 8)^4) - \ln(x^8)$

We can combine these using the logarithmic subtraction rule:

$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$

Thus, the expression becomes:

$\ln\left(\frac{(x + 8)^4}{x^8}\right)$

Final Answer:

$\ln\left(\frac{(x + 8)^4}{x^8}\right)$

This is the condensed logarithmic expression.

Would you like more details on the process, or have any other questions? Here are a few related questions:

1. What other logarithmic properties are useful in condensing expressions?
2. How can you simplify a logarithmic expression further?
3. What would happen if you had negative values inside the logarithms?
4. How do you evaluate logarithmic expressions when specific values for $x$ are given?
5. What does the power rule for logarithms look like when applied to bases other than $e$?

Tip: When condensing logarithmic expressions, always try to apply the power, product, and quotient rules in the correct order to simplify as much as possible!