Problem 1 - Calculus Early Transcendentals

Question

What are the domain and range of \(\ln x ?\)

Step-by-Step Solution

Answer

Answer: The domain of \(\ln x\) is (0, \infty) and the range is (-\infty, \infty).

Step 1: Understanding the natural logarithm function

The natural logarithm function, denoted as \(\ln x\), is the inverse of the exponential function with base \(e\), \(e^x\). In other words, if \(y = \ln x\), then \(x = e^y\).

Step 2: Find the domain of \(\ln x\)

The domain of a function is the set of all possible inputs (x-values) for which the function is defined. Since the natural logarithm function is the inverse of the exponential function, it is not defined for negative values of \(x\) or when \(x=0\) because \(e^x\) can never be equal to a negative number or 0. Therefore, the domain of \(\ln x\) is all positive real numbers: Domain: \((0, \infty)\)

Step 3: Find the range of \(\ln x\)

The range of a function is the set of all possible output values (y-values) for which the function is defined. Given that the function \(y = \ln x\) is the inverse of \(x = e^y\), it is important to analyze how the exponential function behaves. Since \(e^y\) can take on any positive value as \(y\) varies over the real numbers, \(\ln x\) covers all real numbers as its output values. Therefore, the range of \(\ln x\) is all real numbers: Range: \((-\infty, \infty)\)