expand the logarithm. assume all expressions exist and are w write your answer as a sum or difference of…

Explanation:

Step1: Recall the logarithm product rule

The product rule for logarithms states that (\log_b(ac) = \log_b(a) + \log_b(c)) for any positive real numbers (a), (c), and (b>0), (b\neq1). We can apply this rule repeatedly when there are more than two factors in the argument.

Step2: Apply the product rule to (\log_v(xwu))

The argument (xwu) is a product of three factors: (x), (w), and (u). First, we can group (x) and (wu) (or any two factors) and apply the product rule. Let's start by considering (xwu = x \cdot (wu)). Then, by the product rule: (\log_v(x \cdot (wu)) = \log_v(x) + \log_v(wu))

Step3: Apply the product rule again to (\log_v(wu))

Now, we apply the product rule to (\log_v(wu)) since (wu = w \cdot u). Using the product rule: (\log_v(wu) = \log_v(w) + \log_v(u))

Step4: Combine the results

Substituting (\log_v(wu) = \log_v(w) + \log_v(u)) back into the equation from Step 2, we get: (\log_v(xwu) = \log_v(x) + \log_v(w) + \log_v(u))

Answer:

(\log_v x + \log_v w + \log_v u)