To solve the expression \(\log_3 \left(\sqrt{\cos(x) \cdot (x^5 - x)}\right)\), let's break it down step by step.

### Step 1: Simplify the expression inside the logarithm.
The expression inside the logarithm is:
\[
\sqrt{\cos(x) \cdot (x^5 - x)}
\]

### Step 2: Express the square root as a power.
We can rewrite the square root as a power of \( \frac{1}{2} \):
\[
\sqrt{\cos(x) \cdot (x^5 - x)} = \left[\cos(x) \cdot (x^5 - x)\right]^{\frac{1}{2}}
\]

### Step 3: Apply the logarithm rule.
Using the logarithm rule \(\log_b(a^c) = c \cdot \log_b(a)\), we can pull the exponent outside:
\[
\log_3 \left(\left[\cos(x) \cdot (x^5 - x)\right]^{\frac{1}{2}}\right) = \frac{1}{2} \cdot \log_3\left(\cos(x) \cdot (x^5 - x)\right)
\]

### Step 4: Apply the logarithm product rule.
Using the product rule for logarithms \(\log_b(a \cdot c) = \log_b(a) + \log_b(c)\), we can expand the logarithm:
\[
\frac{1}{2} \cdot \log_3\left(\cos(x) \cdot (x^5 - x)\right) = \frac{1}{2} \cdot \left[\log_3(\cos(x)) + \log_3(x^5 - x)\right]
\]

### Final Answer:
The simplified expression is:
\[
\frac{1}{2} \cdot \left[\log_3(\cos(x)) + \log_3(x^5 - x)\right]
\]

This is the final form of the expression.

Would you like any further details on this solution?

### Related Questions:
1. How does the logarithm product rule work?
2. What is the significance of changing a square root to a power of \( \frac{1}{2} \)?
3. Can you explain more about logarithm base change formulas?
4. How would this expression change if the base of the logarithm was different?
5. How do you solve logarithmic equations involving trigonometric functions?

### Tip:
When simplifying logarithmic expressions, always remember to use the logarithm rules for powers, products, and quotients effectively.

Learn how to simplify the logarithmic expression log3(sqrt(cos(x) * (x^5 - x))) step by step. This solution covers key concepts such as logarithmic rules, trigonometry, and algebra, suitable for students in Grades 10-12.

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