Let's solve the given problem using the properties of logarithms.

The expression is:
$\frac{1}{2} \log_3(x) - 2 \log_3(y) - \log_3(z)$

Step-by-Step Solution:

Step 1: Apply the power rule of logarithms

The power rule states: $a \cdot \log_b(M) = \log_b(M^a)$ Using this rule: $\frac{1}{2} \log_3(x) = \log_3\left(x^{\frac{1}{2}}\right) = \log_3\left(\sqrt{x}\right)$ $2 \log_3(y) = \log_3\left(y^2\right)$

Now, the expression becomes: $\log_3\left(\sqrt{x}\right) - \log_3\left(y^2\right) - \log_3(z)$

Step 2: Apply the difference rule of logarithms

The difference rule states: $\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)$ Applying this rule to the first two terms: $\log_3\left(\frac{\sqrt{x}}{y^2}\right) - \log_3(z)$

Step 3: Apply the difference rule again

Now apply the difference rule to the entire expression: $\log_3\left(\frac{\frac{\sqrt{x}}{y^2}}{z}\right) = \log_3\left(\frac{\sqrt{x}}{y^2 z}\right)$

Final Answer:

$\log_3\left(\frac{\sqrt{x}}{y^2 z}\right)$

This is the expression written as a single logarithm.

Would you like further clarification or more details on any of the steps?

Related Questions:

1. What are the other logarithm rules, such as the product rule?
2. How do you handle expressions with different logarithmic bases?
3. What happens if the logarithmic base is not specified?
4. How does changing the base of a logarithm work?
5. Can you apply these logarithmic rules in solving equations?

Tip: When combining logarithms, always check if coefficients can first be converted into exponents using the power rule.