To express [tex]\(\log _3(\sqrt[5]{x} \cdot y)\)[/tex] as a sum or difference, we need to use the properties of logarithms. Let's break this down step-by-step:
1. Identify the expression inside the logarithm:
[tex]\(\log_3(\sqrt[5]{x} \cdot y)\)[/tex]
2. Utilize the properties of logarithms:
- The product rule for logarithms: [tex]\(\log_b(a \cdot c) = \log_b(a) + \log_b(c)\)[/tex]
- The power rule for logarithms: [tex]\(\log_b(a^n) = n \cdot \log_b(a)\)[/tex]
3. First, handle the product inside the logarithm:
[tex]\[
\log_3(\sqrt[5]{x} \cdot y) = \log_3(\sqrt[5]{x}) + \log_3(y)
\][/tex]
4. Deal with the root in the first logarithm term:
The root [tex]\(\sqrt[5]{x}\)[/tex] can be rewritten as [tex]\(x^{1/5}\)[/tex], so:
[tex]\[
\log_3(\sqrt[5]{x}) = \log_3(x^{1/5})
\][/tex]
5. Apply the power rule to the first term:
[tex]\[
\log_3(x^{1/5}) = \frac{1}{5} \log_3(x)
\][/tex]
6. Combine the results:
[tex]\[
\log_3(\sqrt[5]{x} \cdot y) = \frac{1}{5} \log_3(x) + \log_3(y)
\][/tex]
Therefore, the correct answer is:
A. [tex]\(\frac{1}{5} \log_3 x + \log_3 y\)[/tex]
