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Logarithm expansion involves breaking down a complex logarithmic expression into simpler, more manageable parts using various logarithmic properties. This process can transform an initially daunting logarithmic expression into a series of basic logarithms that can be more easily interpreted and, where possible, evaluated without the use of a calculator.

For example, in the given exercise, logarithm expansion takes the form of separating the initial log of a quotient into the difference of two logarithms, and further breaking down those logs into sums of logs for individual components. This technique is particularly useful when dealing with expressions that involve products, quotients, and exponents within a logarithm.

Logarithmic expressions are representations of log functions that specify the relationship between two proportionally related quantities. The expression \(\log_b a\) answers the question: 'To what exponent must the base \(b\) be raised in order to yield \(a\)?' Understanding this fundamental concept allows us to manipulate complex logarithmic expressions using properties of logarithms in order to simplify them or solve equations.

Each property of logarithms relates back to the definitions of logarithms and their inverse relationship with exponentials. Mastery of these expressions and their properties is vital to working effectively across various mathematical fields, including algebra, calculus and beyond.

The quotient rule is one of the essential properties of logarithms, stating that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. Expressed mathematically, this property is written as \(\log\frac{a}{b} = \log a - \log b\).

This powerful property not only aids in the expansion of logarithmic expressions but also streamlines the process of solving logarithmic equations involving division. Using the quotient rule correctly is critical for avoiding common mistakes, such as forgetting to distribute the logarithm to each term inside the parenthesis or mistaking the rule for a different property.

The product rule for logarithms asserts that the logarithm of a product is equivalent to the sum of logarithms for each factor. This is formally expressed as \(\log(ab) = \log a + \log b\).

Applying the product rule systematically can simplify complex logarithmic expressions containing multiple multiplying variables and constants. It is a stepping stone towards expanding log expressions, which ultimately can make it easier to either evaluate the log without a calculator or solve for specific variables. A common usage of this rule, as seen in the exercise's Step 2, enables us to split the logarithm of a multiplied expression into separate, more manageable parts.

The power rule is an essential property that deals with logarithms of exponential expressions. It states that the logarithm of a number raised to an exponent is equal to the exponent multiplied by the logarithm of the number itself. In equation form, \(\log(a^b) = b\cdot\log(a)\).

This rule lends a hand when simplifying log expressions with exponents, such as \(\(\log x^{3}\)\) becoming \(\(3\log x\)\) as shown in Step 3 of the provided solution. Correct application of the power rule can considerably simplify the process of expanding logarithmic expressions and solving logarithmic equations that would otherwise be much more complicated to handle.