The expression given is:

$0.5 \log 0.5 + 0.5 \log 0.5$

First, we can simplify the terms.

$0.5 \log 0.5 + 0.5 \log 0.5 = 2 \times (0.5 \log 0.5)$

Now, simplifying further:

$2 \times (0.5 \log 0.5) = \log 0.5$

We know that:

$\log 0.5 = \log \left( \frac{1}{2} \right) = -\log 2$

Thus, the simplified result is:

$\log 0.5 = -\log 2$

Therefore, the final answer is:

$-\log 2$

Would you like more details or have any questions?

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Here are 5 related questions to expand on this topic:

1. What are the properties of logarithms and how do they help simplify expressions?
2. How would this expression change if it involved a natural logarithm instead?
3. What is the value of $\log 2$ and how is it used in different contexts?
4. How can logarithmic expressions like these be applied in solving exponential equations?
5. What is the relationship between $\log$ and $\ln$, and how are they used differently?

Tip: Always remember that $\log \frac{1}{x} = -\log x$, a useful property when dealing with fractions in logarithmic expressions.