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Logarithmic functions are the inverse operations of exponentiation. They are usually expressed in the form of \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm and \( x \) is the argument. The function is defined only when the argument \( x \) is a positive real number (\( x > 0 \)), as you cannot take the logarithm of a negative number or zero.

It's important to remember that the base \( b \) must also be a positive number, and it cannot be 1 because the logarithm would be undefined. When the base is the special number \( e \) (approximately 2.71828), which is the base of the natural logarithm, the function is denoted by \( f(x) = \ln(x) \). The natural logarithm has unique properties that make it particularly useful in various areas of mathematics and its applications.

When we're dealing with inequalities, such as finding the domain of a logarithmic function, we often need to solve an inequality to determine the set of numbers that can be substituted into the function. An inequality states that one expression is less than or greater than another. The solutions to an inequality can sometimes be a range of values, forming an interval. For example, \( x < 7 \) means that \( x \) can be any number that is less than 7.

To solve an inequality, we perform similar operations to solving equalities, but we must keep in mind one important rule: whenever we multiply or divide both sides of the inequality by a negative number, we have to flip the inequality sign. This rule is essential to obtain the correct solution set for inequalities.

The domain of a function is the complete set of possible values of the independent variable—in simpler terms, it's the collection of all the input values for which the function is defined. For logarithmic functions, the domain comprises all real numbers that result in the argument of the logarithm being positive. Graphically, you can visualize the domain as the projection of the function curve onto the x-axis.

Finding the domain involves understanding the constraints or limitations of the function. For instance, in the given exercise, \( f(x) = \log(7-x) \), the argument \( (7-x) \) must be greater than 0. Hence the domain converts to the inequality \( x < 7 \), meaning all real numbers less than 7 can be plugged into the function. Recognizing the conditions for a valid domain is a crucial step not only in solving problems involving logarithms but also in studying the behavior of all types of mathematical functions.