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The power rule of logarithms is a fundamental property used to manipulate logarithmic expressions. It states that for any positive real numbers \(a\) and \(b\), and any real number \(c\), the expression \(\log_{a}(b^c)\) can be rewritten as \(c \, \log_{a}(b)\). This rule is very useful when dealing with exponents within logarithms. For example, in the expression \( \log_{m} \left( \frac{5r^3}{z^5} \right)^{1/2} \), the exponent \(\frac{1}{2}\) can be moved in front of the logarithm, simplifying the expression: \( \frac{1}{2} \, \log_{m} \left( \frac{5r^3}{z^5} \right) \). This makes it easier to apply other logarithmic rules.

The quotient rule of logarithms helps simplify the logarithms of fractions. It states that for any positive real numbers \(a\), \(b\), and \(c\), \( \log_a \left( \frac{b}{c} \right) \) can be simplified to \( \log_a(b) - \log_a(c) \). This means the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. For instance, in our expression \( \frac{1}{2} \log_m \left( \frac{5r^3}{z^5} \right) \), applying the quotient rule gives us \( \frac{1}{2} (\log_m (5r^3) - \log_m (z^5)) \). This allows each part of the fraction to be handled separately, which simplifies the computation further.

The product rule of logarithms is another key property. It states that for any positive real numbers \(a\), \(b\), and \(c\), \( \log_a(bc) \) can be rewritten as \( \log_a(b) + \log_a(c) \). That is, the logarithm of a product is the sum of the logarithms of the factors. Using this rule in our expression, we can simplify \(\log_m (5r^3)\) as \(\log_m (5) + \log_m (r^3)\). So our expression becomes \( \frac{1}{2} (\log_m (5) + \log_m (r^3) - \log_m (z^5)) \). This further breaks down complex expressions into simpler parts.

Simplifying logarithmic expressions often requires using a combination of logarithm properties like the power, quotient, and product rules. For example, take \( \frac{1}{2} (\log_m 5 + \log_m r^3 - \log_m z^5) \). Notice that \( \log_m r^3 \) and \( \log_m z^5 \) can both be simplified further using the power rule. Applying this rule, we transform each term: \( \log_m r^3 \) becomes \( 3 \log_m r \) and \( \log_m z^5 \) becomes \( 5 \log_m z \). Substituting back, we get: \( \frac{1}{2} (\log_m 5 + 3 \log_m r - 5 \log_m z) \). Finally, distribute \( \frac{1}{2} \): \( \frac{1}{2} \log_m 5 + \frac{3}{2} \log_m r - \frac{5}{2} \log_m z \). This combination of rules streamlines complex logarithmic expressions into simpler, more manageable parts.