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, ∞), and the range of \(\ln{x}\) is (-∞, ∞).

Step 1: Understanding the natural logarithm function

The natural logarithm function, denoted as \(\ln{x}\), is the inverse function of the exponential function \(e^x\). In other words, if we have a function \(y=e^x\), then the natural logarithm function of y is defined as \(\ln y =x\). The natural logarithm function is defined for positive real numbers and is undefined for negative numbers and zero.

Step 2: Determining the domain of \(\ln{x}\)

As mentioned in Step 1, the natural logarithm function is defined only for positive real numbers. Therefore, the domain of \(\ln{x}\) consists of all positive real numbers. In interval notation, this can be represented as: Domain of \(\ln{x} = (0, \infty)\)

Step 3: Finding the range of \(\ln{x}\)

Now that we have determined the domain, let's find the range of the natural logarithm function. Since the natural logarithm is the inverse of the exponential function, it "undoes" the exponential function. The exponential function \(e^x\) can yield any positive real number for any given value of x, and hence its range is \((0, \infty)\). Since \(\ln{x}\) is the inverse of the exponential function, its range must cover all real numbers x for which the exponential function is defined. Therefore, the range of \(\ln{x}\) is: Range of \(\ln{x} = (-\infty, \infty)\) In conclusion: - The domain of \(\ln{x}\) is \((0, \infty)\) - The range of \(\ln{x}\) is \((-\infty, \infty)\)