Problem 102 - Intermediate Algebra

Question

In the next section we will show that the logarithmic function \(f(x)=\log _{2} x\) is the inverse of the exponential function \(f(x)=2^{x}\). From that information, how could you sketch a graph of \(f(x)=\log _{2} x\) ?

Step-by-Step Solution

Answer

Reflect the graph of \(2^{x}\) across the line \(y=x\) to sketch \(\log_{2} x\).

Step 1: Understand the Relationship

The logarithmic function \(f(x) = \log_{2} x\) is the inverse of the exponential function \(f(x) = 2^{x}\). This means that for any \(y = 2^{x}\), the opposite transformation is \(x = \log_{2} y\).

Step 2: Sketch the Exponential Function

Graph the exponential function \(g(x) = 2^x\). It passes through the point (0,1) because \(2^{0} = 1\), and it grows rapidly as \(x\) increases. The curve approaches the x-axis but never touches it, creating a horizontal asymptote at \(y = 0\).

Step 3: Reflect Across the Line \(y = x\)

Since \(f(x) = \log_{2} x\) is the inverse of \(g(x) = 2^x\), reflect the graph of \(g(x)\) across the line \(y = x\). The identity line \(y = x\) helps visualize this transformation.

Step 4: Sketch \(f(x) = \log_{2} x\)

Draw the reflected graph from Step 3, which passes through the point (1,0) and has a vertical asymptote at \(x = 0\). The curve increases slowly as \(x\) grows and reflects the rapid increase of \(g(x) = 2^x\).

Step 5: Verify Key Features

Check that the graph includes the point (1,0), similar to how the exponential graph included (0,1). Ensure the curve approaches the y-axis (\(x = 0\)) asymptotically, confirming the reflection is executed correctly.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. In simple terms, they help us find the exponent. If you know the logarithm, you can determine what power we must raise a base to get a certain number. For example, if we have the logarithmic equation \(f(x) = \log_{2} x\), what we are essentially saying is: "What power must we raise 2 to get \(x\)?"
This function is crucial for solving equations where the unknown variable is an exponent. Here are some key points to remember:

• \(\log_{b}(x)\) is the inverse of \(b^{x}\). If you do one operation, you can reverse it by applying the other.
• It includes a vertical asymptote at \(x = 0\), which means the graph approaches this line but never touches or crosses it.
• It passes through the point (1,0), signifying that the logarithm of 1 is always zero, regardless of the base.

Understanding logarithmic functions is essential as they are the mirror image of their exponential counterparts.

Exponential Functions

Exponential functions involve numbers raised to a variable power, such as \(g(x) = 2^x\). When graphed, they quickly grow, moving upwards as \(x\) increases. A few things to keep in mind while dealing with exponential functions include:

• They have a horizontal asymptote at \(y=0\), meaning the graph will get infinitely close to the x-axis but forbears to touch it.
• The function passes through the point (0,1), which is a direct result of raising any non-zero number to the zero power.
• Exponential functions showcase rapid change. This is why they are used to model situations with quick growth, like population increase or compound interest.

The correlation between exponential functions and their inverses, logarithmic functions, is one of the pivotal aspects of understanding them.

Graph Sketching

Graph sketching with logarithmic and exponential functions can reveal insights visually. The key method is to use reflections. When sketching the graph of a logarithmic function, you can follow these steps:

• First, draw the exponential function. For example, \(g(x) = 2^x\) starts at (0,1) and ascends sharply.
• Next, reflect this graph across the line \(y = x\). This line helps you flip the existing graph onto its inverse.
• The resulting graph represents the logarithmic function, in this case \(f(x) = \log_{2} x\), crossing at (1,0) and approaching \(x=0\) asymptotically.

Using reflections helps us see the relationship between the functions and fosters a deeper understanding by visualizing the connection.