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 \(ln(x)\) is \(y=e^x\). Its domain is \(\mathbb{R}\) (all real numbers) and its range is \((0, +\infty)\) (all positive real numbers).

Step 1: Understand the problem

We are given the natural logarithm function, \(\ln(x)\), and we want to find its inverse function and determine its domain and range.

Step 2: Recall the properties of logarithmic functions

One important property of logarithmic functions is that the logarithm is the inverse operation of the exponential function. That is, if \(y=\ln(x)\), then \(x=e^y\).

Step 3: Find the inverse function

Since \(\ln(x)\) is the natural logarithm, its inverse function is the exponential function with base \(e\). To find the inverse of \(\ln(x),\) we can switch the variables \(x\) and \(y\) in the equation \(y=\ln(x)\) and then solve for \(y\). Doing this, we have: \(x=\ln{(y)}\) Now, to solve for \(y\), we can use the property mentioned in Step 2: \(y=e^x\) The inverse function of \(\ln(x)\) is \(y=e^x\).

Step 4: Determine the domain and range of the inverse function

The domain of \(e^x\) is the set of all real numbers, because the exponential function is defined for all real values of \(x\). So, the domain of the inverse function is \(\mathbb{R}\). The range of \(e^x\) is the set of all positive real numbers since the exponential function always returns a positive value. In other words, the range of the inverse function is \((0, +\infty)\).

Step 5: Summary

The inverse function of \(\ln(x)\) is \(y=e^x\). The domain of this inverse function is \(\mathbb{R}\), and its range is \((0, +\infty)\).