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

Verified step by step guidance

1. Recall that the domain of a logarithmic function \( f(x) = \ln(g(x)) \) requires the argument \( g(x) \) to be strictly greater than zero, so we need to find where \( (x-2)^2 > 0 \).
2. Set up the inequality \( (x-2)^2 > 0 \) and analyze it. Since a square of any real number is always non-negative, \( (x-2)^2 \geq 0 \) for all \( x \), but it equals zero when \( x = 2 \).
3. Because the logarithm is undefined at zero, exclude \( x = 2 \) from the domain. Therefore, the domain includes all real numbers except \( x = 2 \).
4. Express the domain in interval notation as \( (-\infty, 2) \cup (2, \infty) \).
5. Summarize that the domain of \( f(x) = \ln((x-2)^2) \) is all real numbers except \( x = 2 \), because the argument of the logarithm must be positive.

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 to values that make the argument inside the logarithm positive, since the logarithm of zero or a negative number is undefined.

Properties of Logarithmic Functions

A logarithmic function, such as ln(x), is only defined for positive arguments. This means the expression inside the logarithm must be greater than zero. Understanding this property is essential to determine the domain by setting the argument greater than zero and solving the inequality.

Solving Inequalities Involving Squares

When the argument of the logarithm is a squared expression, like (x-2)², it is always non-negative. To find the domain, you must analyze when the squared expression is strictly greater than zero, since the logarithm requires a positive argument, not zero. This involves solving inequalities and understanding the behavior of squared terms.