Problem 17 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

The given function is the natural logarithm function, denoted as \(f(x) = \ln x\). This function represents the natural logarithm (to the base \(e\)) of \(x\).

Step 2: Recall the Properties of Logarithms

The natural logarithm function is the inverse of the exponential function with base \(e\). This means, for \(f(x) = \ln x\), there exists a function \(g(x)\) such that \(f(g(x)) = x\) and \(g(f(x)) = x\).

Step 3: Defining the Inverse Function

To find the inverse function of \(f(x) = \ln x\), we need to express \(y = \ln x\) in terms of \(x\). This involves solving for \(x\) in terms of \(y\).

Step 4: Solving for the Inverse Function

Starting from the equation \(y = \ln x\), we want to solve for \(x\). By converting the logarithmic equation into its exponential form, we have \(x = e^y\).

Step 5: Writing the Inverse Function

Now that we have \(x = e^y\), we can express the inverse function. This gives us \(y = f^{-1}(x) = e^x\). Therefore, the inverse of \(f(x) = \ln x\) is \(f^{-1}(x) = e^x\).

Key Concepts

Natural Logarithm

The natural logarithm is a key concept in algebra and calculus. It is denoted as \( \ln x \) and represents the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. This special number \( e \) is known as Euler's number, and it is a fundamental constant in mathematics.

The natural logarithm has the following important properties:

• \( \ln(1) = 0 \): because \( e^0 = 1 \)
• \( \ln(e) = 1 \): since \( e^1 = e \)
• \( \ln(xy) = \ln x + \ln y \)
• \( \ln\left(\frac{x}{y}\right) = \ln x - \ln y \)

These properties make the natural logarithm especially useful in solving problems involving exponential functions.

Understanding the natural logarithm is essential to comprehend other mathematical processes, such as finding an inverse function, which relies on converting logarithmic equations to exponential ones.

Inverse Function

The concept of an inverse function is fundamental in mathematics. Essentially, an inverse function 'undoes' what the original function does. If you have a function \( f(x) \), the inverse function, denoted as \( f^{-1}(x) \), will reverse the operation of \( f(x) \).

To determine the inverse of a function, follow these steps:

• Start with the function equation \( y = f(x) \).
• Switch the roles of \( x \) and \( y \), so the equation becomes \( x = f(y) \).
• Solve for \( y \) in terms of \( x \) to find the inverse function \( f^{-1}(x) \).

For example, with \( f(x) = \ln x \), swapping \( x \) and \( y \) gives you \( x = \ln y \). Solving for \( y \), you transform this into its exponential form, \( y = e^x \), which is the inverse function \( f^{-1}(x) = e^x \).

Inverse functions are valuable because they allow us to reverse equations, expressing output values back in terms of their original input.

Exponential Function

Exponential functions are a cornerstone in algebra. These functions are defined by having a constant as their base raised to a variable exponent. The exponential function with base \( e \) is depicted as \( e^x \), where \( e \) is Euler's number.

The exponential function \( y = e^x \) has properties that make it powerful and versatile:

• It always yields positive results since \( e^x > 0 \) for all real \( x \).
• It is the inverse of the natural logarithm function, meaning \( \ln(e^x) = x \) and \( e^{\ln x} = x \).
• The derivative of \( e^x \) is itself, \( \frac{d}{dx}e^x = e^x \), which makes it unique in calculus.

Understanding exponential functions is crucial, as they are applied in diverse fields ranging from microbiology to finance.

In the context of inverse functions, the exponential function often appears as the solution when converting natural logarithms back to their original form. For instance, the inverse of \( f(x) = \ln x \) is \( f^{-1}(x) = e^x \), showcasing the intrinsic link between logarithms and exponentials.