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Algebraic modeling is a method used to represent real world situations with algebraic expressions or equations. Taking our exercise for example, the growth in height of a girl as she ages is represented using the function

\[f(x)=62+35 \log (x-4)\]

Here, age is the variable that we change, and the function gives us the corresponding percentage of her adult height attained. Algebraic models like these turn empirical data and observed patterns into mathematical formats that can be analyzed and used to make predictions. Effective models require understanding the underlying principles and accurately representing relationships between quantities with mathematical symbols and operations.

Logarithmic expressions are the inverse operations of exponentiation and have countless applications in real-world contexts, including the modeling of growth and decay. In the function given in our exercise,

\[\log (x-4)\]

shows up within the model. The use of the logarithm here is integral for the equation, that models the non-linear pattern of growth in height. Students need to comprehend how to work with logarithms, including understanding their properties, how to simplify them, and using a calculator to find their numerical values. In basic algebra, the logarithm base 10 is commonly used, and this is implied when no base is explicitly stated.

Percentages are another frequently encountered concept in algebra. They describe how one quantity relates to another in terms of hundredths. In our exercise, we are asked to find the percentage of the adult height of a girl at a certain age. The equation given translates the age (minus 4) into a percentage. When dealing with percentages in algebra, it's important to remember that we are often looking for a part of a whole, and the equation or function in use should reflect that relationship. Comprehending how percentages function within algebraic expressions allows students to solve a range of problems, from those involving growth, like in our exercise, to financial calculations among many other applications.

Mathematical functions are paramount to representing relationships between quantities in a precise and predictable way. A function takes an input, processes it through a formula, and produces an output. In our case, the function

\[f(x)=62+35 \log (x-4)\]

is a specific type of function known as a logarithmic function, reflecting a pattern in the growth of a girl's height. Understanding functions involves recognizing the types (linear, quadratic, exponential, etc.), properties such as domain and range, and how alterations to the function's equation can affect its graph and outputs. Including the concept of a function within algebraic operations is essential, as it aids in mapping out numerous types of real-world situations quantitatively.