Question

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

Short answer

Answer: The domain of \(\ln x\) is \((0, \infty)\), and the range of \(\ln x\) is the set of all real numbers.

Step-by-step solution

Understand the natural logarithm function

The natural logarithm function, \(\ln x\), is the inverse function of \(y = e^x\), where \(e\) is Euler's number (\(\approx 2.71828\)). For a function to have an inverse, it must be both one-to-one and onto. Therefore, we will analyze the exponential function \(y = e^x\) first and then find the inverse of that function to determine the domain and range of \(\ln x\).

Analyze the exponential function \(y = e^x\)

The exponential function \(y = e^x\) is a continuous and strictly increasing function for all real numbers. This means that the domain of \(y = e^x\) is the set of all real numbers, and the range of \(y = e^x\) is the set of all positive real numbers, \((0, \infty)\) (since \(e^x\) is always positive for any real value of x).

Identify domain and range of \(\ln x\)

Since \(\ln x\) is the inverse function of \(y = e^x\), we can interchange the domain and range of the exponential function to find the domain and range for the natural logarithm function. The domain of \(e^x\) is the set of all real numbers, which is the range of \(\ln x\). The range of \(e^x\) is the set of all positive real numbers, \((0, \infty)\), which is the domain of \(\ln x\). Therefore, the domain of \(\ln x\) is \((0, \infty)\), and the range of \(\ln x\) is the set of all real numbers.

Key concepts

Inverse Function

An inverse function is a function that "undoes" the action of another function. If you have a function, say \( f(x) \), its inverse, denoted \( f^{-1}(x) \), will return the original input when applied to the output of \( f(x) \). In simpler terms, if you apply a function and then its inverse, you'll end up where you started.
Essentially, inverse functions reverse each other's processes.

• For a function to have an inverse, it must be one-to-one; each input corresponds to exactly one unique output.
• It must also be onto, meaning that every element of the range is mapped for some input.

Understanding inverse functions is crucial because they allow us to solve equations and understand relationships between quantities. In the case of the logarithm and exponential functions, they exhibit a special relationship where each is the inverse of the other.

Exponential Function

The exponential function, particularly the natural exponential function \( y = e^x \), is a fundamental mathematical function where the constant \( e \) (Euler's number) is raised to the power of \( x \). This function is critical in group of mathematical continuous functions, and is known for its unique properties.

• It is always continuous and smooth, with a constant rate of growth.
• It is defined for all real numbers, which means its domain is the set of all real numbers \(( -\infty, \infty)\).
• Its range, on the other hand, only includes positive real numbers \((0, \infty)\), as \( e^x \) is always positive.

The exponential function grows exceptionally fast, and its inverse is the natural logarithm \( \ln x \). Understanding both the behavior and properties of \( y = e^x \) is essential in calculus, differential equations, and any domain involving exponential growth or decay.

Euler's Number

Euler's number, known as \( e \), is an irrational number approximately equal to 2.71828. It is named after the Swiss mathematician Leonhard Euler. This surprising yet important constant appears naturally in many areas of mathematics, especially in continuous growth and calculus.
Here are some key aspects of \( e \):

• It is the base of the natural logarithm, which means \( \ln(e) = 1 \).
• The number \( e \) can be defined as the sum of an infinite series, \( e = \sum_{n=0}^{\infty} \frac{1}{n!} \).
• It serves as the base for natural exponential functions, which describe processes such as compound interest, population growth, and radioactive decay.

The properties of Euler's number make it invaluable in both theoretical and applied mathematics. Its presence in exponential and logarithmic functions highlight why understanding \( e \) is essential in the broader context of mathematical functions.