91Ó°ÊÓ

Understanding the quotient rule of logarithms is essential when dealing with division within a logarithmic expression. This rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In mathematical terms, this is expressed as
\[ \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \].
The core benefit of this rule is its ability to simplify complex logarithmic expressions by breaking them down into more manageable parts. For example, given the expression \( \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) \), you can apply the quotient rule to obtain \( \log_{b}(x^2y) - \log_{b}(z^2) \). This simplification process facilitates easier computation and further manipulation of the logarithmic terms, as seen in the exercise provided.

Moving forward with the product rule of logarithms, this rule empowers you to dissect logarithms of products into sums of logarithms. Formally, it states that
\[ \log_b(xy) = \log_b(x) + \log_b(y) \].
Applying the product rule simplifies multiplication within a logarithm, breaking it down into additive components. Taking the intermediate step from the exercise, \( \log_{b}(x^2y) \), by employing the product rule, we end up with \( \log_{b}{x^2} + \log_{b}{y} \). This division into individual logarithms of the factors makes the expression more straightforward, allowing for the application of additional properties, such as the power rule, which further refines the complexity of the expression. The utilization of the product rule is a keystone in expanding logarithmic expressions to a form that can be easily interpreted or computed.

The power rule of logarithms is a swift way to handle exponents within logarithmic expressions. This powerful rule declares that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the base number. Mathematically, it's presented as
\[ \log_b(x^k) = k \cdot \log_b(x) \].
In the context of our exercise, applying the power rule transforms \( \log_{b}{x^2} \) and \( \log_{b}{z^2} \) into \( 2\log_{b}x \) and \( -2\log_{b}z \) respectively. By doing so, the exponents become coefficients, significantly simplifying the original logarithmic expression. The rule is particularly useful when inversing the process of exponentiation in a logarithm, offering a direct path to reducing complex expressions into more elementary terms. Embracing this rule enhances understanding and solving exercises involving logarithms with precision and ease.