The given expression is:

$\log \left[ x (x + 5) (x + 3)^4 \right], \quad x > 0$

We can use the properties of logarithms to simplify it. These properties are:

1. $\log(ab) = \log(a) + \log(b)$
2. $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$
3. $\log(a^n) = n \log(a)$

Step 1: Apply the product rule

Using the first property, the expression can be split into a sum of logarithms:

$\log \left[ x (x + 5) (x + 3)^4 \right] = \log(x) + \log(x + 5) + \log\left((x + 3)^4\right)$

Step 2: Apply the power rule

Now, apply the third property to the term $\log\left((x + 3)^4\right)$, bringing the exponent 4 to the front:

$\log(x) + \log(x + 5) + 4 \log(x + 3)$

Final Answer:

$\log(x) + \log(x + 5) + 4 \log(x + 3)$

This is the expression written as a sum of logarithms.

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Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the expression change if the logarithm had a base other than 10?
2. Can you apply these logarithmic properties to an expression involving division?
3. What happens if the expression inside the logarithm contains a square root?
4. How do you handle logarithmic expressions where the base is unknown or variable?
5. Can you solve equations where logarithmic expressions are set equal to a constant?

Tip: Always remember the basic properties of logarithms—they're key to simplifying complex expressions!