Express in index form: log$_a^x$ + log$_a^y$ = 3

Analyze the question: The expression is log_a x + log_a y = 3. We need to rewrite it in index form, meaning in the form a^n = N.

Concept:

1. Product law of logarithms: log_a M + log_a N = log_a (MN)
2. Relationship between logarithm and index forms: log_a N = k means N = a^k

Solution:

Step 1: Combine the logs using the product law. log_a x + log_a y = log_a (xy)

Step 2: Substitute into the equation. log_a (xy) = 3

Step 3: Convert to index form. xy = a³

Checking the other options:

• A and D wrongly add x and y. The addition law of logs gives a product, not a sum.
• B ignores the base: it treats 3 as the value of xy, but 3 is the power of the base a.

Answer: The correct option is C. xy = a³.

Tip: When you see two logs with the same base being added, think "multiply the insides." Then use the "base to the power of the answer" rule to turn the log into an index: log_a N = k → N = a^k.