What are the domain and range of ln x?

Verified step by step guidance

1. Understand the function: The natural logarithm function, ln(x), is defined as the logarithm to the base e, where e is approximately 2.718. It is only defined for positive values of x.
2. Determine the domain: The domain of ln(x) consists of all x-values for which the function is defined. Since ln(x) is undefined for x ≤ 0, the domain is (0, ∞).
3. Analyze the behavior of ln(x): As x approaches 0 from the right (x → 0⁺), ln(x) decreases without bound, approaching negative infinity. As x increases (x → ∞), ln(x) increases without bound.
4. Determine the range: Based on the behavior of ln(x), the range includes all real numbers because the function can output any value from negative infinity to positive infinity. Thus, the range is (-∞, ∞).
5. Summarize the domain and range: The domain of ln(x) is (0, ∞), and the range is (-∞, ∞). These properties are fundamental to understanding the natural logarithm function.

Key Concepts

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is defined for positive real numbers and is the inverse of the exponential function e^x. Understanding this function is crucial for determining its domain and range.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the natural logarithm function ln(x), the domain is limited to positive real numbers, meaning x must be greater than zero, as ln(x) is undefined for x ≤ 0.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For the natural logarithm function ln(x), the range is all real numbers, as ln(x) can take any value from negative infinity to positive infinity as x approaches zero from the right and increases without bound.