Find the domain of each logarithmic function. f(x) = log (2 - x)

Verified step by step guidance

1. Recall that the domain of a logarithmic function \( f(x) = \log_b(g(x)) \) requires the argument \( g(x) \) to be greater than zero, because the logarithm of zero or a negative number is undefined.
2. Identify the argument of the logarithm in the function \( f(x) = \log(2 - x) \). Here, the argument is \( 2 - x \).
3. Set up the inequality to find the domain: \( 2 - x > 0 \). This inequality ensures the argument is positive.
4. Solve the inequality \( 2 - x > 0 \) by isolating \( x \). Subtract 2 from both sides to get \( -x > -2 \), then multiply both sides by \( -1 \) (remember to reverse the inequality sign) to get \( x < 2 \).
5. Conclude that the domain of \( f(x) = \log(2 - x) \) is all real numbers \( x \) such that \( x < 2 \).

Key Concepts

Domain of a Function

The domain of a function is the set of all input values (x-values) for which the function is defined. For logarithmic functions, the domain is restricted because the argument inside the log must be positive. Identifying the domain involves finding all x-values that make the expression inside the log greater than zero.

Properties of Logarithmic Functions

A logarithmic function, such as f(x) = log(g(x)), is only defined when its argument g(x) is positive. This means that for f(x) = log(2 - x), the expression 2 - x must be greater than zero. Understanding this property is essential to determine the domain of the function.

Inequalities and Solving for Domain

To find the domain of f(x) = log(2 - x), you solve the inequality 2 - x > 0. This involves algebraic manipulation to isolate x and express the domain in interval notation. Mastery of solving inequalities is crucial for correctly identifying the domain of logarithmic functions.