To write the expression as a single logarithm, let's analyze it. The given expression is:

$\log_a(x) + 2\log_a(y) - \frac{1}{2}\log_a(z)$

We'll use logarithmic properties to combine these terms:

1. Logarithm of a product: $\log_a(x) + \log_a(y) = \log_a(xy)$
2. Logarithm of a power: $k \log_a(x) = \log_a(x^k)$
3. Logarithm of a quotient: $\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$

Step 1: Apply the power rule

• $2\log_a(y) = \log_a(y^2)$
• $\frac{1}{2}\log_a(z) = \log_a(z^{1/2}) = \log_a(\sqrt{z})$

This simplifies the expression to: $\log_a(x) + \log_a(y^2) - \log_a(\sqrt{z})$

Step 2: Apply the product and quotient rules

$\log_a\left(x \cdot y^2\right) - \log_a(\sqrt{z}) = \log_a\left(\frac{x y^2}{\sqrt{z}}\right)$

Final Result

The expression as a single logarithm is: $\log_a\left(\frac{x y^2}{\sqrt{z}}\right)$

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Would you like further explanations or details on logarithmic properties?

Here are 5 related questions to deepen your understanding:

1. How do you expand a single logarithm into multiple terms?
2. What is the relationship between logarithms and exponents?
3. How do you simplify logarithmic expressions with fractional exponents?
4. Can you express $\log_a\left(\frac{a^3}{b^2}\right)$ as a sum or difference of logs?
5. How does the base of the logarithm affect simplification?

Tip: Always remember to check the domain of the variables when simplifying logarithmic expressions to ensure they are valid.