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A logarithmic expression is an equation that involves the logarithm of a number. In general, it takes the form \( \log_{b}(x) \), where \( b \) is the base of the logarithm, and \( x \) is the argument or the number you're taking the log of. Logarithms are essentially the inverse of exponentiation. While exponentiation tells you how to go from the base to a number, logarithms tell you which power you raised the base to get that number.

Using logarithmic expressions, we can solve equations that have exponential growth or decay. They're widely used in many fields such as science, engineering, and financial mathematics to model phenomena that grow or shrink exponentially over time. Understanding the basics of a logarithmic expression is crucial for grasping more complex properties of logarithms.

When we talk about the expanded form in logarithms, we're referring to breaking down a complex logarithmic expression into simpler, separate logarithms. This is accomplished using different logarithmic properties. For example, consider the expression \( \log _{m}\left(\frac{67}{83}\right) \). In its current form, the logarithm is applied to a fraction. Our goal is to simplify or "expand" this expression.

To expand this, we use the properties of logarithms. By using that property, we can transform it into:

• \( \log_{m}(67) - \log_{m}(83) \)

Breaking it down like this helps in simplifying computations, solving equations, and understanding the relationship between the components of the original fraction.

The logarithmic property of division is a fundamental rule that helps simplify the logs of fractions. This property states: \( \log_{b}\left(\frac{x}{y}\right) = \log_{b}(x) - \log_{b}(y) \). It illustrates how a logarithm of a quotient can be rewritten as a difference of logarithms. By transforming a single logarithmic expression into a subtraction between two simpler ones, this property is exceptionally useful, especially in algebraic manipulations and calculus where precision and simplification are key.

Let's consider the example \( \log_{m}\left(\frac{67}{83}\right) \) from our original problem. Using this property, we can break it down as:

• \( \log_{m}(67) - \log_{m}(83) \)

This makes it clear how each part of the initial fraction is represented separately in the expanded form. It's a valuable tool in both simplifying complex logarithmic equations and aiding in the comprehension of logarithmic relationships in a more digestible format.