Question

which is equivalent to \\(\log(x^2 y)\\)?
a \\(\log(2x) + \log(y)\\)
b \\(\log(2x) - \log(y)\\)
c \\(2\log(x) + \log(y)\\)
d \\(2\log(x) - \log(y)\\)

Explanation:

Step1: Recall Logarithm Product Rule

The logarithm of a product \( \log(ab) = \log(a) + \log(b) \) and the power rule \( \log(a^n)=n\log(a) \).

Step2: Apply Rules to \( \log(x^2y) \)

First, use the product rule: \( \log(x^2y)=\log(x^2)+\log(y) \).
Then, apply the power rule to \( \log(x^2) \): \( \log(x^2) = 2\log(x) \).
So, \( \log(x^2y)=2\log(x)+\log(y) \).

Answer:

C. \( 2\log(x) + \log(y) \)