To solve $\log x + \log y - 2 \log z$ as a single logarithm, let's apply the properties of logarithms:

1. Addition Rule: $\log a + \log b = \log (a \cdot b)$.
2. Multiplication by a Constant: $c \log a = \log (a^c)$.

Step-by-Step Solution

1. Combine $\log x + \log y$: $\log x + \log y = \log (x \cdot y) = \log (xy)$

2. Apply the Constant Rule to $2 \log z$: $2 \log z = \log (z^2)$

3. Combine the Results Using Subtraction: $\log (xy) - \log (z^2) = \log \left( \frac{xy}{z^2} \right)$

Thus, the expression $\log x + \log y - 2 \log z$ simplifies to: $\log \left( \frac{xy}{z^2} \right)$

Answer

The correct answer is: $\log \frac{xy}{z^2}$

This corresponds to the third option in the provided choices.

Would you like further details, or do you have any questions?

Here are 5 questions to deepen your understanding:

1. How would the expression change if it was $\log x + 2\log y - \log z$?
2. What happens if we have $\log x - \log y + 3\log z$ instead?
3. Why does $c \log a = \log (a^c)$ work for logarithms?
4. Can you convert $\log \frac{a}{b} + \log \frac{c}{d}$ into a single logarithm?
5. How does the base of the logarithm (e.g., base 10 or base $e$) affect such simplifications?

Tip: Remember, breaking down logarithmic expressions often involves converting addition into multiplication and subtraction into division.