Question

1. patrick expanded $2\log_{3}x^{3}w^{5}$ to $6\log_{3}x + 5\log_{a}w$. determine if he is correct. explain in complete sentences

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 \(\log_3w\).