Problem 17 - Intermediate Algebra

Question

What is the inverse of the natural logarithmic function \(f(x)=\ln x ?\)

Step-by-Step Solution

Answer

The inverse function of \( f(x) = \ln x \) is \( f^{-1}(x) = e^x \).

Step 1: Understand the Function

The natural logarithmic function is given by \( f(x) = \ln x \). Our objective is to find the inverse of this function. The inverse function, \( f^{-1}(x) \), should satisfy \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Step 2: Swap and Solve for x

To find the inverse, we start by swapping \( x \) and \( y \) in the equation \( y = \ln x \), which becomes \( x = \ln y \). We need to solve this equation for \( y \) in terms of \( x \).

Step 3: Exponentiate Both Sides

Recognizing that taking the exponential of a logarithm will enable us to isolate \( y \), we apply the exponential function to both sides: \( e^x = e^{\ln y} \).

Step 4: Simplify to Find y

Since \(e^{\ln y} = y\) by the property of logarithms, it follows that \( y = e^x \). Thus, the inverse function is \( f^{-1}(x) = e^x \).

Step 5: Verify the Inverse

Verify by checking if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \). For \( f(f^{-1}(x)) = f(e^x) = \ln(e^x) = x \), and \( f^{-1}(f(x)) = f^{-1}(\ln x) = e^{\ln x} = x \), confirming that \( f^{-1}(x) = e^x \) is indeed the inverse of \( f(x) = \ln x \).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a special type of logarithm. Its base is the constant \( e \), approximately 2.71828. This constant \( e \) is crucial in many areas of mathematics due to its unique properties. The natural logarithm is the inverse of the exponential function with base \( e \). Hence, if you know \( y = \ln x \), it implies that \( x = e^y \).

Natural logarithms have several important properties. Here are a few key ones:

• The natural logarithm of 1 is 0, since \( \ln(1) = 0 \).
• Logarithm rules apply, such as \( \ln(ab) = \ln a + \ln b \).
• We also have \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).
• The natural logarithm increases slowly compared to regular linear growth, tending towards infinity as \( x \) approaches infinity.

Recognizing these properties helps in simplifying expressions involving natural logs.

When solving equations involving natural logarithms, you might often use its inverse nature with the exponential to switch formats. This is key when finding inverses of natural logarithmic functions.

Exponential Functions

Exponential functions describe processes that increase or decrease at a rate proportional to their current value. The mathematical form of an exponential function is \( f(x) = a \cdot e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithm.

The simplest form of an exponential function is \( e^x \), which is the inverse of the natural logarithm. This function grows exceptionally fast and is unique because its rate of growth is proportional to its size. In other words, the derivative of \( e^x \) is also \( e^x \).

Some important properties of exponential functions include:

• \( e^0 = 1 \)
• \( e^{x+y} = e^x \cdot e^y \)
• \( e^{-x} = \frac{1}{e^x} \)

These properties make exponential functions powerful tools for modeling growth, compound interest, and natural phenomena such as radioactive decay. Understanding these basic rules is essential for solving equations where exponential functions play a critical role, such as in the inverse of \( \ln x \).

Function Properties

Functions have key properties that help us understand their behavior and relationship with other functions. Understanding these properties can be particularly valuable when dealing with inverse functions like the natural logarithm and its inverse, the exponential function.

• Inverse Functions: When you have a function \( f \), its inverse \( f^{-1} \) undoes the action of \( f \). For instance, since \( f(x) = \ln x \), its inverse is \( f^{-1}(x) = e^x \), because \( \ln(e^x) = x \) and \( e^{\ln x} = x \). The original function and its inverse essentially 'cancel' each other out.
• Domain and Range: The domain of the natural logarithm function \( \ln x \) is \( x > 0 \), and the range is all real numbers. For the inverse \( f^{-1}(x) = e^x \), the domain is all real numbers and the range is \( y > 0 \). Observing domain and range swap in inverse functions is an important concept.
• Composition of Functions: When two functions are inverses of each other, composing them results in the identity function, \( f(f^{-1}(x)) = f^{-1}(f(x)) = x \). This confirms the idea that inverses cancel each other out effectively.

Understanding these properties helps in deeper analysis and verification of both functions and their inverses, ensuring correct calculations and interpretations in mathematical problems.