6) patrick expanded $2\\log_{3}x^{3}w^{5}$ to $6\\log_{3}x + 5\\log_{a}w$. determine if he is correct…

Explanation:

Step1: Apply Product Rule of Logarithms

The product rule of logarithms states that (\log_b(MN)=\log_bM + \log_bN). So, for (\log_3(x^3w^5)), we can expand it as (\log_3x^3+\log_3w^5).

Step2: Apply Power Rule of Logarithms

The power rule of logarithms states that (\log_bM^n = n\log_bM). Applying this to (\log_3x^3) and (\log_3w^5), we get (3\log_3x) and (5\log_3w) respectively. So, (\log_3(x^3w^5)=3\log_3x + 5\log_3w).

Step3: Multiply by 2

Now, we have the original expression (2\log_3(x^3w^5)). Multiply the expanded form of (\log_3(x^3w^5)) by 2: (2(3\log_3x + 5\log_3w)=6\log_3x+10\log_3w).

Step4: Compare with Patrick's Answer

Patrick's answer is (6\log_3x + 5\log_aw) (note the base of the second logarithm is (a) instead of (3) and the coefficient is (5) instead of (10)). So, Patrick made two mistakes: the base of the second logarithm is incorrect (should be (3) not (a)) and the coefficient of (\log_3w) should be (10) not (5).

Answer:

Patrick is not correct. When expanding (2\log_3x^3w^5), first use the product rule (\log_b(MN)=\log_bM+\log_bN) to get (2(\log_3x^3 + \log_3w^5)). Then use the power rule (\log_bM^n=n\log_bM) to get (2(3\log_3x + 5\log_3w)), which simplifies to (6\log_3x+10\log_3w). Patrick's answer has an incorrect base ((a) instead of (3)) for the second logarithm and an incorrect coefficient ((5) instead of (10)) for (\log_3w).