Question

what is $2\log x - \log y - 2\log z$ written as a single logarithm?\
\
$\circ$ $\log \frac{x^2}{yz^2}$\
\
$\circ$ $\log \frac{2x}{2yz}$\
\
$\circ$ $\log \frac{x^2 y}{z^2}$\
\
$\circ$ $\log \frac{(xz)^2}{y}$

Explanation:

Step1: Apply Power Rule

Use the power rule of logarithms: \( n\log a=\log a^n \).
For \( 2\log x \), we get \( \log x^2 \).
For \( 2\log z \), we get \( \log z^2 \).
So the expression becomes \( \log x^2 - \log y - \log z^2 \).

Step2: Apply Quotient Rule

The quotient rule of logarithms is \( \log a - \log b=\log \frac{a}{b} \).
First, combine \( \log x^2 - \log y \): \( \log \frac{x^2}{y} \).
Then, combine \( \log \frac{x^2}{y} - \log z^2 \): \( \log \frac{\frac{x^2}{y}}{z^2} \).
Simplify \( \frac{\frac{x^2}{y}}{z^2}=\frac{x^2}{yz^2} \).
Thus, the expression is \( \log \frac{x^2}{yz^2} \).

Answer:

A. \( \log \frac{x^2}{yz^2} \)