Question

expand the following logarithmic expression into a sum or difference of logs. \\(\log_{4}(xy)\\) \\(\bigcirc\\) \\(\log_{4}(x) + \log_{4}(y)\\) \\(\bigcirc\\) \\(\frac{\log_{4}(x)}{\log_{4}(y)}\\) \\(\bigcirc\\) \\(\log_{4}(x) - \log_{4}(y)\\) \\(\bigcirc\\) \\(\log_{4}(x)\log_{4}(y)\\) \\(\bigcirc\\) none of the above

Explanation:

Step1: Recall Logarithm Product Rule

The product rule for logarithms states that for any positive numbers \(a\), \(b\), and base \(c>0, c
eq1\), \(\log_{c}(ab)=\log_{c}(a)+\log_{c}(b)\).

Step2: Apply the Rule to Given Expression

Here, the expression is \(\log_{4}(xy)\), where \(a = x\), \(b = y\), and \(c = 4\). Using the product rule, we get \(\log_{4}(xy)=\log_{4}(x)+\log_{4}(y)\).

Answer:

A. \(\log_{4}(x) + \log_{4}(y)\)