Certainly! Let's condense the expression \(\log_9 z + \frac{\log_9 x}{3} + \frac{\log_9 y}{3}\) into a single logarithm.
1. Combine the logarithms using the properties of logarithms:
The expression is \(\log_9 z + \frac{\log_9 x}{3} + \frac{\log_9 y}{3}\).
First, recall the property of logarithms that states \(\frac{\log_b a^c}{d} = \frac{\log_b a}{d} = \log_b a^{\frac{c}{d}}\).
Applying this to \(\frac{\log_9 x}{3}\):
[tex]\[
\frac{\log_9 x}{3} = \log_9 x^{\frac{1}{3}}
\][/tex]
Applying this to \(\frac{\log_9 y}{3}\):
[tex]\[
\frac{\log_9 y}{3} = \log_9 y^{\frac{1}{3}}
\][/tex]
So the expression becomes:
[tex]\[
\log_9 z + \log_9 x^{\frac{1}{3}} + \log_9 y^{\frac{1}{3}}
\][/tex]
2. Combine the logarithms using the product rule:
The product rule for logarithms states that \(\log_b a + \log_b c = \log_b (a \cdot c)\).
Applying this to the expression:
[tex]\[
\log_9 z + \log_9 x^{\frac{1}{3}} = \log_9 (z \cdot x^{\frac{1}{3}})
\][/tex]
Then further including \(\log_9 y^{\frac{1}{3}}\):
[tex]\[
\log_9 (z \cdot x^{\frac{1}{3}}) + \log_9 y^{\frac{1}{3}} = \log_9 (z \cdot x^{\frac{1}{3}} \cdot y^{\frac{1}{3}})
\][/tex]
3. Simplify the expression:
Combining \(x^{\frac{1}{3}}\) and \(y^{\frac{1}{3}}\):
[tex]\[
z \cdot x^{\frac{1}{3}} \cdot y^{\frac{1}{3}} = z \cdot (x \cdot y)^{\frac{1}{3}}
\][/tex]
So the condensed expression in terms of a single logarithm is:
[tex]\[
\log_9 \left(z \cdot (x \cdot y)^{\frac{1}{3}}\right)
\][/tex]
Given the choices provided, the most accurate answer is:
[tex]\[
\log_9 \left(z \cdot \sqrt[3]{x \cdot y}\right)
\][/tex]
This matches with one of the given answer choices:
[tex]\[
\log_9(z \cdot \sqrt[3]{yx})
\][/tex]
