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The given expression is:

\[
\log_2 \left( \frac{\sin(u) - \cos(u) + 3\sqrt{2}}{\sqrt{2}} \right)
\]

This simplifies to:

\[
\frac{\log_2\left( \sqrt{2} \left( \sin(u) - \cos(u) + 3\sqrt{2} \right) \right)}{2}
\]

Would you like me to proceed with a further simplification or explore the solution for specific values of \( u \)?

Here are some related questions:

1. What is the range of the expression \( \sin(u) - \cos(u) \)?
2. How does the presence of \( \sqrt{2} \) affect the logarithmic function?
3. What are the key properties of logarithms when applied to trigonometric expressions?
4. How does the logarithmic base (base 2) impact the result?
5. Can this expression be simplified further for specific values of \( u \)?

**Tip:** Remember that logarithms of square roots can often be simplified by using properties of exponents and logarithms.

Solve the expression: log_2((sin(u) - cos(u) + 3√2) / √2)

Explore how to simplify the trigonometric logarithmic expression log_2((sin(u) - cos(u) + 3√2) / √2). This problem involves key mathematical concepts like trigonometry and logarithmic properties, making it ideal for high school students in Grades 11-12.

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