To rewrite [tex]\(2 \log x - \log y - 2 \log z\)[/tex] as a single logarithm, let's use the properties of logarithms. Here is a detailed, step-by-step solution:
1. Apply the Power Rule of Logarithms:
- The power rule states that [tex]\( a \log b = \log b^a \)[/tex].
- Apply this to [tex]\(2 \log x\)[/tex]: [tex]\(2 \log x = \log x^2\)[/tex].
- Apply this to [tex]\(2 \log z\)[/tex]: [tex]\(2 \log z = \log z^2\)[/tex].
So, the expression becomes:
[tex]\[ \log x^2 - \log y - \log z^2 \][/tex]
2. Apply the Quotient Rule of Logarithms:
- The quotient rule states that [tex]\(\log a - \log b = \log \left(\frac{a}{b}\right)\)[/tex].
- Combine [tex]\(\log x^2 - \log y\)[/tex] using the quotient rule: [tex]\(\log x^2 - \log y = \log \left(\frac{x^2}{y}\right)\)[/tex].
Now we have:
[tex]\[ \log \left(\frac{x^2}{y}\right) - \log z^2 \][/tex]
3. Apply the Quotient Rule of Logarithms Again:
- Combine [tex]\(\log \left(\frac{x^2}{y}\right) - \log z^2\)[/tex] using the quotient rule: [tex]\(\log \left(\frac{x^2}{y} \right) - \log z^2 = \log \left(\frac{\frac{x^2}{y}}{z^2}\right) = \log \left(\frac{x^2}{y z^2}\right)\)[/tex].
4. Final Simplified Form:
- The expression [tex]\(2 \log x - \log y - 2 \log z\)[/tex] simplified is [tex]\(\log \left(\frac{x^2}{y z^2}\right)\)[/tex].
Therefore, the correct option is:
[tex]\(\log \frac{x^2}{y z^2}\)[/tex].
