Problem 42 - Precalculus Mathematics for Calculus

Question

Sketch the graph of the function by plotting points. $$g(x)=\log _{4} x$$

Step-by-Step Solution

Answer

Plot points (1, 0), (4, 1), and (16, 2) for \(g(x) = \log_4 x\) to sketch the curve, noting the vertical asymptote at \(x = 0\).

Step 1: Understanding the Function

We need to sketch the graph of the function \(g(x) = \log_4 x\). This is a logarithmic function with base 4, which means it is the inverse of the exponential function \(f(x) = 4^x\). The graph will be a curve passing through certain key points, which we will plot to sketch the function.

Step 2: Selecting Key Points

For a logarithmic function, it's useful to calculate specific points where the x values are powers of the base. Let's select some points: \(x = 1, 4, \) and \(16\). These points will help us understand the behavior of the graph.

Step 3: Calculating Values

Calculate \(g(x)\) for each selected \(x\):- \(x = 1\): \(g(1) = \log_4 1 = 0\), because \(4^0 = 1\).- \(x = 4\): \(g(4) = \log_4 4 = 1\), because \(4^1 = 4\).- \(x = 16\): \(g(16) = \log_4 16 = 2\), because \(4^2 = 16\).Plot these points as (1, 0), (4, 1), and (16, 2) on the graph.

Step 4: Sketching the Graph

Draw a smooth curve passing through the points you have plotted. The graph of \(g(x) = \log_4 x\) starts very close to the y-axis and increases slowly, taking a steeper curve upward after the point (4, 1). The graph approaches the y-axis but never touches it, indicating a vertical asymptote at \(x = 0\).

Step 5: Analyzing the Graph Behavior

The graph has several features:- It passes through the point (1, 0) since \(\log_4 1 = 0\).- It increases and passes through (4, 1) and (16, 2).- The graph has a vertical asymptote at \(x = 0\) because the logarithm is undefined for \(x \leq 0\).- It is always increasing but at a decreasing rate.

Key Concepts

Logarithmic Function

A logarithmic function is a powerful mathematical concept often used in different fields like science, engineering, and economics. The function we are discussing here is \(g(x) = \log_4 x\), which is a logarithm with the base 4. This means that it answers the question: "To what power must 4 be raised, to yield \(x\)?" Logarithmic functions have a unique property where they transform multiplication into addition, simplifying many complex calculations. They are crucial for solving exponential equations and understanding growth and decay processes.

Instead of growing steadily like a line, logarithmic functions grow rapidly at first and then slow down. As \(x\) increases, the rate of growth of \(g(x)\) decreases. The function passes through key points such as (1,0), (4,1), and (16,2), showing a pattern as it curves upwards but becomes less steep as \(x\) gets larger.

Inverse of Exponential Function

The logarithmic function is essentially the inverse of an exponential function. For our example, \(g(x) = \log_4 x\), this relationship means that if \(g(x)\) outputs a value \(y\), then the exponentiation \(4^y = x\) holds true. In simpler terms, exponentiation and logarithms undo each other. This property allows us to understand complex exponentials by converting them into more manageable logarithmic equations.
The inverse nature also means that where the exponential function \(f(x) = 4^x\) is increasing exponentially, the logarithmic function grows increasingly slowly. An exponential curve shoots up rapidly, whereas the logarithmic curve rises sluggishly over time. This makes logarithms excellent tools for dealing with problems involving expansion or decay in a controlled manner.

Vertical Asymptote

Vertical asymptotes are crucial in understanding the behavior of graphs of logarithmic functions. In the function \(g(x) = \log_4 x\), there is a vertical asymptote at \(x = 0\). This means that as \(x\) approaches 0 from the positive side, the value \(g(x)\) descends toward negative infinity. The graph gets infinitely close to the y-axis but never actually meets it.

This is because the logarithm of zero is undefined, and no real number can serve as an answer to "4 raised to what power gives zero?" This inability to achieve or even approach zero assures that the graph will climb up along the y-axis without ever reaching it. Recognizing vertical asymptotes helps in predicting the abrupt changes in values and can guide us in understanding the other characteristics of the function.

Plotting Points

Plotting points is key in sketching the graph of any function, including logarithmic ones. In our function \(g(x) = \log_4 x\), selecting helpful points involves choosing x-values that are powers of 4, such as 1, 4, and 16. These points simplify calculations and offer a clear scale for plotting. Once we plug in these x-values, we calculate \(g(x)\) to get corresponding y-values.

• For \(x=1\), \(g(x)=0\)
• For \(x=4\), \(g(x)=1\)
• For \(x=16\), \(g(x)=2\)

By plotting these points—(1,0), (4,1), and (16,2)—on a coordinate plane, a clearer image of the graph emerges. As you draw a smooth curve through these points, you can visualize how the graph behaves as it approaches the vertical asymptote and stretches upward. Using plotted points to guide the sketch ensures accuracy and aids in conveying the function's true shape.