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The given expression simplifies to:

\[
0.5 \cdot \log(x^4) + \frac{1}{3} \cdot \log((x + 2)^3) - 0.5 \cdot \log((x^2 - x - 6)^2)
\]

This can be further simplified using logarithmic properties:

1. \(\log(a^b) = b \cdot \log(a)\)
2. \(\log(a) - \log(b) = \log\left(\frac{a}{b}\right)\)

Applying these:

\[
\log(x^4) = 4 \cdot \log(x), \quad \log((x + 2)^3) = 3 \cdot \log(x + 2), \quad \log((x^2 - x - 6)^2) = 2 \cdot \log(x^2 - x - 6)
\]

Thus, the expression simplifies further to:

\[
2 \cdot \log(x) + \log(x + 2) - \log(x^2 - x - 6)
\]

Finally:

\[
\log\left(\frac{2x(x + 2)}{x^2 - x - 6}\right)
\]

Would you like further clarification or details on this? 

Here are 5 related questions you could explore:
1. How do logarithmic properties apply to simplify complex expressions?
2. What are the conditions for the argument inside a logarithm to be valid?
3. How can this simplified logarithmic expression be used in solving equations?
4. What are the common rules for manipulating expressions involving logarithms?
5. How do we solve logarithmic equations that involve multiple variables?

**Tip:** Always check the domain of the logarithmic function by ensuring that the arguments are positive.

1/3 * log((x + 2)^3) + 1/2 * [log(x^4) - log((x^2 - x - 6)^2)]

Learn how to simplify the expression 1/3 * log((x + 2)^3) + 1/2 * [log(x^4) - log((x^2 - x - 6)^2)] using logarithmic properties such as log(a^b) = b * log(a) and log(a) - log(b) = log(a/b). This problem demonstrates the application of these properties for solving complex logarithmic expressions.

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