Problem 110 - Elementary and Intermediate Algebra

Question

How are the functions \(f(x)=\ln x\) and \(g(x)=e^{x}\) related?

Step-by-Step Solution

Answer

The functions \(f(x)=\ln x\) and \(g(x)=e^x\) are inverses of each other.

Step 1: Understanding the Functions

The two functions we are dealing with are the natural logarithm function, \(f(x) = \ln x\), and the exponential function, \(g(x) = e^x\). The natural logarithm is the inverse of the exponential function with the base \(e\). This means the operations of \(f(x)\) and \(g(x)\) can cancel each other out under certain conditions.

Step 2: Inverse Relationships

Since \(f(x) = \ln x\) and \(g(x) = e^x\) are inverses, applying \(f\) to \(g(x)\) or \(g\) to \(f(x)\) yields the identity function over their domains. Specifically, \(\ln(e^x) = x\) for all real numbers \(x\), and \(e^{\ln x} = x\) for all \(x > 0\). This inverse relationship is the key way these functions relate.

Step 3: Checking the Inversion

To confirm their relationship, let's compute \(f(g(x))\) and \(g(f(x))\):1. \(f(g(x)) = \ln(e^x) = x\), confirming that \(f\) is the inverse of \(g\) when considering positive \(x\).2. \(g(f(x)) = e^{\ln x} = x\) when \(x > 0\), confirming that \(g\) is the inverse of \(f\) when \(x\) is within its domain (\(x > 0\)).

Step 4: Conclusion

These functions are mathematical inverses of each other. When composed, \(f(g(x))\) results in \(x\) and \(g(f(x))\) also leads to \(x\) within their respective domains, showcasing how they cancel each other's actions.

Key Concepts

Exponential Function

An exponential function is a type of mathematical function that has the form \(g(x) = a^x\), where \(a\) is a constant and \(x\) is an exponent. Among the most common types of exponential functions is \(g(x) = e^x\), where \(e\) is a special number approximately equal to 2.718281828. It's known as the base of the natural logarithm.
Exponential functions are very significant in mathematics due to their unique properties:

• They grow rapidly; for example, as \(x\) increases, \(e^x\) increases quickly.
• They always output positive values since \(a^x > 0\) for any real number \(x\).
• They have a horizontal asymptote at \(y = 0\), meaning the graph gets closer and closer to this line but never actually touches it for \(x\) going to negative infinity.

Understanding exponential functions is crucial due to their applications in sciences, such as modeling population growth and radioactive decay.
Because \(g(x) = e^x\) is one of the most popular, it's essential to comprehend its relationship with the natural logarithm function.

Natural Logarithm

The natural logarithm function is written as \(f(x) = \ln x\) and is the inverse of the exponential function \(g(x) = e^x\). The natural logarithm has a base of \(e\) and is vital in mathematics because it transforms multiplication into addition, which simplifies complex mathematical operations.
Several key characteristics of the natural logarithm include:

• It's only defined for positive numbers, which means \(x > 0\).
• It has a very interesting graph: the function is continuous and increases slowly as \(x\) increases.
• It crosses the y-axis at one point: when \(x = 1\), because \(\ln(1) = 0\).

The power of the natural logarithm lies in its ability to "undo" the actions of the exponential function. When you apply \(\ln\) to an exponential function, for example \(\ln(e^x)\), you simply end up with \(x\).
This fundamental property of being an inverse is what makes the natural logarithm so important in higher-level math.

Function Composition

Function composition is a concept where two functions are used together, applying one function to the result of another. When you compose functions, you create a new function through their combination. If you have two functions, \(f(x)\) and \(g(x)\), the composition \(f(g(x))\) means you first apply \(g\) to \(x\) then \(f\) to the result of \(g(x)\).
In the context of the natural logarithm and exponential functions, when you compose these inversely related functions, something special happens:

• \(f(g(x)) = \ln(e^x) = x\), indicating that applying \(\ln\) after \(e^x\) returns the original \(x\).
• Similarly, \(g(f(x)) = e^{\ln x} = x\) when \(x > 0\), showing the reverse.

This inverse relationship highlights how composing inverses gives you back your starting point.
These operations are the underpinnings of the identity function within their domains. Function composition is a powerful technique, allowing us to interchangeably use exponential functions and logarithms to solve equations and model real-world situations.