Select the correct answer.

How would you write this expression as a sum or difference?
[tex]\log _3(\sqrt[5]{x} \cdot y)[/tex]

A. [tex]\frac{1}{5} \log _3 x+\log _3 y[/tex]

B. [tex]\frac{\log _9 2 \frac{1}{8}}{\log _3 y}[/tex]

C. [tex]\log _3 x^5-\log _3 y[/tex]

D. [tex]\frac{1}{5}\left(\log _3 x+\log _3 y\right)[/tex]



Answer :

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]