To condense the expression $\log x - 2 \log y + 3 \log z$ into the logarithm of a single quantity, we can use the properties of logarithms:

1. Power Rule: $a \log b = \log(b^a)$.
2. Product Rule: $\log a + \log b = \log(ab)$.
3. Quotient Rule: $\log a - \log b = \log\left(\frac{a}{b}\right)$.

Steps:

1. Apply the power rule to $-2 \log y$ and $3 \log z$: $\log x - 2 \log y + 3 \log z = \log x - \log(y^2) + \log(z^3)$

2. Now, use the product and quotient rules: $\log x - \log(y^2) + \log(z^3) = \log \left(\frac{x \cdot z^3}{y^2}\right)$

Final Answer:

$\log \left(\frac{x \cdot z^3}{y^2}\right)$

Would you like a more detailed breakdown or have any questions?

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Here are five related questions to practice:

1. How would you condense $4 \log a + \log b - \frac{1}{2} \log c$?
2. Simplify $\log(x^2) + 3 \log(y) - \log(z)$.
3. Write $\log(a) - \log(b) + 2 \log(c)$ as a single logarithm.
4. If given $2 \log x - 5 \log y + \log z$, how would you condense it?
5. How can you expand $\log\left(\frac{a^2 b}{c^3}\right)$?

Tip: Remember that applying the power rule to convert coefficients is often the first step in simplifying complex logarithmic expressions.