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The power rule of logarithms is a handy tool when dealing with exponents inside a logarithmic expression. This rule states that if you have a logarithm of a number raised to a power, you can bring the exponent out in front of the logarithm. In mathematical terms, if you have an expression like \( \log(a^b) \), it can be rewritten as \( b \cdot \log(a) \). This is incredibly useful because it simplifies the expression and makes it easier to manipulate.Consider the original expression \( \log \sqrt{\frac{x y}{z}} \). The square root can be expressed as a power of 1/2. So, our first step is to rewrite the expression as \( \log \left( \frac{x y}{z} \right)^{1/2} \). Using the power rule, you can take this 1/2 and bring it in front, transforming the expression into \( \frac{1}{2} \cdot \log \left( \frac{x y}{z} \right) \). This simplifies the expression significantly, preparing it for further decomposition using other rules.

Logarithms are often used to separate division terms within a single logarithm into individual parts. This is done using the quotient rule of logarithms. When you have a fraction inside a log, the quotient rule helps rewrite it by separating the numerator and the denominator. In formulaic terms, \( \log \left( \frac{a}{b} \right) = \log(a) - \log(b) \).Given the expression from our previous step \( \frac{1}{2} \cdot \log \left( \frac{x y}{z} \right) \), we can apply the quotient rule. This allows us to separate \( \frac{x y}{z} \) into \( \log(xy) - \log(z) \). Thus, it transforms into \( \frac{1}{2} \cdot (\log(xy) - \log(z)) \). This step helps break down complex logarithmic expressions into manageable parts we can manipulate further.

The product rule of logarithms is about breaking down multiplication inside a logarithm into a sum of individual logarithms. It expresses that for any multiplication inside a logarithm, you can separate it using addition. In formulaic terms, this rule is represented as \( \log(ab) = \log(a) + \log(b) \).In our exercise, after applying the quotient rule, we ended up with \( \log(xy) - \log(z) \). To further simplify \( \log(xy) \), use the product rule: \( \log(xy) = \log(x) + \log(y) \). Applying this, the expression becomes \( \log(x) + \log(y) - \log(z) \). Don't forget the multiplier from earlier—\( \frac{1}{2} \) has to be distributed across this resulting expression.Thus, each term is multiplied by \( \frac{1}{2} \), giving us the final expression: \( \frac{1}{2} \cdot \log(x) + \frac{1}{2} \cdot \log(y) - \frac{1}{2} \cdot \log(z) \). This final result is neatly simplified, showcasing the incredible utility of the product, quotient, and power rules when working with logarithmic expressions.