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Logarithmic expressions are a fundamental part of algebra and are widely used in various fields such as mathematics, physics, and engineering. Essentially, a logarithm answers the question: To what exponent must we raise a certain base to obtain a given number? For example, in the logarithmic expression \( \log_{b}(x) \), the base is \( b \) and \( x \) is the number we are evaluating. When we talk about logarithms, it is important to remember that they are the inverses of exponentials; thus they 'undo' the exponential operation.

When dealing with logarithmic expressions, it's also important to understand the domain restrictions, which typically require the arguments to be positive due to the nature of how logarithms are defined. This is why in the example from the exercise, it was assumed that all variables are positive. Grasping the basics of how logarithms work and their notation is essential for effective problem solving in algebra and beyond.

The quotient rule is one of the properties of logarithms that makes them so versatile and powerful for simplifying expressions. It states that the logarithm of a quotient is equal to the difference of the logarithms. In mathematical terms, \( \log_{b}\left(\frac{a}{c}\right) = \log_{b}(a) - \log_{b}(c) \), where \( b \) is the base and \( a \) and \( c \) are the numerators and denominators of the quotient, respectively. This rule allows us to break down more complex logarithmic expressions, such as the one in our exercise \( \log_{10}\left(\frac{7}{z}\right) \), into simpler parts that are easier to work with.

Understanding the quotient rule is instrumental for expanding, simplifying, and solving logarithmic equations. It also assists in computing logarithms without a calculator, a fundamental skill before the era of digital computation and still relevant today for theoretical understanding and during exams with restricted tool use.

Simplifying logarithms often involves applying various rules, such as the quotient rule as seen in our exercise. Logarithm simplification can lead to more elegant, compact forms of expressions. However, the simplification process is not always straightforward and may not always result in a numerical answer. For instance, in the given exercise, \( \log_{10}(7) \) cannot be simplified any further without resorting to a calculator because 7 is not a power of 10. On the other hand, the \( \log_{10}(z) \) portion of our expression can be simplified by realizing that when dealing with base 10, the notation can be shortened, leading to \( \log(z) \).

This is due to the common logarithm convention, where the base 10 is typically omitted because it is the most frequently used base in logarithms, particularly in many real-world applications like measuring the strength of earthquakes (Richter scale) or acidity (pH scale). Learning to recognize when and how to simplify logarithmic expressions is key to efficiency and accuracy in dealing with logarithmic functions.