Problem 34 - College Algebra with Corequisite Support

Question

For the following exercises, sketch the graphs of each pair of functions on the same axis. \(f(x)=\log (x)\) and \(g(x)=10^{x}\)

Step-by-Step Solution

Answer

Sketch the graph of \(f(x) = \log(x)\) and \(g(x) = 10^x\) on the same axis. \(f(x)\) rises slowly, \(g(x)\) rises quickly.

Step 1: Understanding the Functions

The first function is \( f(x) = \log(x) \), which is the logarithmic function. This function has a vertical asymptote at \( x = 0 \) and is defined for \( x > 0 \). It passes through the point \( (1, 0) \). The second function is \( g(x) = 10^{x} \), which is an exponential function. This function has a horizontal asymptote at \( y = 0 \) and contains the point \( (0, 1) \).

Step 2: Determine the Key Features of Each Function

For \( f(x) = \log(x) \), as \( x \) increases, the function increases slowly. The function approaches negative infinity as \( x \) approaches zero from the right. For \( g(x) = 10^{x} \), as \( x \) increases, the function increases rapidly. As \( x \) decreases to negative infinity, \( g(x) \) approaches zero.

Step 3: Common Points and Intersections

Both functions are continuous for \( x > 0 \). The only intersection point within the domain of both functions is \( x = 1 \), where \( f(1) = 0 \) and \( g(1) = 10 \).

Step 4: Sketching the Graphs

On a coordinate plane, plot the key points: \( (1, 0) \) for \( f(x) \) and \( (0, 1) \), \( (1, 10) \) for \( g(x) \). Draw \( f(x) \), beginning along the vertical asymptote, then transitioning through \( (1, 0) \) and rising slowly. For \( g(x) \), start near the horizontal axis to the left of \( y = 1 \), go through \( (0, 1) \), then rise steeply through \( (1, 10) \).

Step 5: Label and Analyze the Graphs

Label each curve, \( f(x) = \log(x) \) passing through \( (1, 0) \) and continuing to the right slowly increasing. Then label \( g(x) = 10^{x} \), an exponential curve starting at \( (0, 1) \) sharply increasing as it moves right. Note how \( f(x) \) increases more slowly compared to the rapid increase of \( g(x) \) once past the point \( x = 1 \).

Key Concepts

Logarithmic Function

The logarithmic function, expressed as \( f(x) = \log(x) \), is one of the fundamental functions in mathematics. This function is an inverse of the exponential function and is only defined for positive \( x \) values, meaning \( x > 0 \). Let's look at some important features:

• **Vertical Asymptote**: As \( x \) approaches zero from the right, \( f(x) \) moves towards negative infinity. This behavior is due to a vertical asymptote at \( x = 0 \).
• **Growth**: Though it increases as \( x \) becomes larger, the growth of \( f(x) \) is very slow.
• **Intercept**: The graph passes through the point \( (1, 0) \), illustrating that the logarithm of 1 is 0.

Understanding the behavior and restrictions of the logarithmic function is vital when comparing it to other types of functions, like exponential functions.

Exponential Function

Exponential functions, such as \( g(x) = 10^x \), are distinguished by their rapid growth rate. They represent processes where quantities grow or decay at a constant rate per unit interval. Here are key characteristics:

• **Horizontal Asymptote**: As \( x \) moves towards negative infinity, \( g(x) \) approaches zero, indicating a horizontal asymptote at \( y = 0 \).
• **Growth Rate**: Unlike the logarithmic function, \( g(x) \) increases very quickly as \( x \) becomes more significant.
• **Intercept**: It passes through the point \( (0, 1) \), as any number to the power of zero is 1.

This function's rapid increase makes it especially significant in fields such as finance, biology, and physics. Recognizing its graph helps in understanding phenomena that escalate quickly.

Asymptotes

The topic of asymptotes is crucial when graphing functions like logarithmic and exponential functions. An asymptote is a line that a graph approaches but never touches or crosses. Here's how they apply to our functions:

• **Vertical Asymptote in Logarithmic Function**: The graph of \( f(x) = \log(x) \) has a vertical asymptote at \( x = 0 \). This means that as \( x \) gets closer to zero from the positive side, the value of \( f(x) \) decreases without bound (moves toward negative infinity).
• **Horizontal Asymptote in Exponential Function**: Conversely, \( g(x) = 10^x \) has a horizontal asymptote at \( y = 0 \). As \( x \) decreases to negative infinity, \( g(x) \) gets closer and closer to zero but never quite reaches it.

Understanding these asymptotes helps in predicting how functions behave before graphing, particularly their trends near axes.

Intersection Points

Finding the intersection points of functions is essential to understand where graphs overlap. This also provides insights into solutions of certain equations. In our example:

• For \( f(x) = \log(x) \) and \( g(x) = 10^x \), the only intersection within their domain is at \( x = 1 \).
• At this point, \( f(1) = 0 \) and \( g(1) = 10 \), which means the functions both evaluate at a significant point in terms of position on the axis but at different heights (y-values).

Identifying and understanding intersection points is useful in various applications like solving equations and determining the comparative growth rates between different functions.