91Ó°ÊÓ

The logarithm quotient rule is a fundamental property of logarithms. It helps us deal with the division of terms inside a logarithm. The rule states: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This means that the logarithm of a quotient is the difference of the logarithms. For our exercise, applying this rule helps separate the numerator from the denominator. We then look at each part individually. The numerator \( \sqrt{x} \cdot \sqrt[3]{y} \) and the denominator \( w^2 \sqrt{z} \) each become subjects of their own logarithms.

Next, let's use the logarithm product rule. This rule simplifies expressions where terms are multiplied inside a logarithm. The rule is given by: \( \log_b(M \cdot N) = \log_b M + \log_b N \). Using this in our context allows us to break down the terms \( \sqrt{x} \cdot \sqrt[3]{y} \) and \( w^2 \sqrt{z} \) into separate logarithms. This simplifies our complex expression into simpler parts. For instance, \( \log_3(\sqrt{x} \cdot \sqrt[3]{y}) = \log_3 \sqrt{x} + \log_3 \sqrt[3]{y} \). Similarly, \( \log_3(w^2 \sqrt{z}) = \log_3 w^2 + \log_3 \sqrt{z} \). Now we have isolated the terms, making it easier to work with them.

Finally, we use the logarithm power rule to further simplify. This rule states: \( \log_b(M^p) = p\log_b M \). It lets us move any exponent in front of the logarithm. Applying this to our terms, we see: \( \log_3 \sqrt{x} = \log_3(x^{1/2}) = \frac{1}{2} \log_3 x \), \( \log_3 \sqrt[3]{y} = \log_3(y^{1/3}) = \frac{1}{3}\log_3 y \), \( \log_3 w^2 = 2 \log_3 w \) and \( \log_3 \sqrt{z} = \log_3(z^{1/2}) = \frac{1}{2} \log_3 z \). Combining all results, we simplify our original expression to: \( \log_3 \left( \frac{\sqrt{x} \cdot \sqrt[3]{y}}{w^2 \sqrt{z}} \right) = \frac{1}{2} \log_3 x + \frac{1}{3}\log_3 y - 2 \log_3 w - \frac{1}{2} \log_3 z \). This demonstrates the power and utility of these logarithm properties when simplifying complex expressions.