Problem 4 - Calculus Early Transcendentals

Question

What is the inverse function of \(\ln x,\) and what are its domain and range?

Step-by-Step Solution

Answer

Answer: The inverse function of the natural logarithm, \(\ln x\), is \(y = e^{x}\), with a domain of \((-\infty, \infty)\) and a range of \((0, \infty)\).

Step 1: Identify the Original Function and Its Inverse Relationship

The original function is the natural logarithm, denoted as \(\ln x\). The inverse function is the one that reverses the natural logarithm operation, which is the exponential function with base \(e\), denoted as \(e^{x}\).

Step 2: Express the Original Function and Inverse Function

Let \(y\) be the original function: \(y = \ln x\) Now, switch the roles of \(x\) and \(y\) to find the inverse function: \(x = \ln y\)

Step 3: Solve for the Inverse Function

To solve for the inverse function, we need to exponentiate both sides of the equation using base \(e\). It will cancel out the natural logarithm: \(e^{x} = e^{\ln y}\) Since \(e^{\ln y}\) is equal to \(y\), we get the inverse function: \(y = e^{x}\)

Step 4: Find the Domain and Range of the Original Function

The domain of the original function, \(\ln x\), is the set of real numbers for which the natural logarithm is defined. The logarithm is only defined for positive numbers, so the domain of \(\ln x\) is: Domain of \(\ln x\): \((0, \infty)\) The range of the original function, \(\ln x\), is the set of real numbers to which the logarithm can map. Since the logarithm can output any real number, the range of \(\ln x\) is: Range of \(\ln x\): \((-\infty, \infty)\)

Step 5: Find the Domain and Range of the Inverse Function

For the inverse function \(y = e^{x}\), the domain and range are reversed from the original function. The domain of \(e^{x}\) is the set of real numbers to which the exponential function is defined. Since the exponential function can accept any real number as an input, the domain of \(e^x\) is: Domain of \(e^{x}\): \((-\infty, \infty)\) The range of the inverse function \(y = e^{x}\) is the set of real numbers to which the exponential function can map. The exponential function can only output positive numbers, so the range of \(e^{x}\) is: Range of \(e^{x}\): \((0, \infty)\) In conclusion, the inverse function of \(\ln x\) is \(y = e^{x}\), with a domain of \((-\infty, \infty)\) and a range of \((0, \infty)\).