Problem 36 - College Algebra

Question

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

Step-by-Step Solution

Answer

Graph \( f(x) = \log(x) \) starting just right of \( x=0 \); graph \( g(x) = 10^{x} \) from a low point upward.

Step 1: Identify the Functions

We have two functions to graph: \( f(x) = \log(x) \) and \( g(x) = 10^{x} \). We'll sketch these functions on the same set of axes to compare them.

Step 2: Understand Logarithmic Function

The function \( f(x) = \log(x) \) represents the logarithm of \( x \) to base 10, also known as the common logarithm. This function is only defined for \( x > 0 \). It passes through the point (1,0) since \( \log(1) = 0 \), and it approaches negative infinity as \( x \) approaches zero.

Step 3: Understand Exponential Function

The function \( g(x) = 10^{x} \) is an exponential function with base 10. It is defined for all real numbers and passes through the point (0,1) since \( 10^0 = 1 \). As \( x \) increases, the value of \( g(x) \) grows very rapidly.

Step 4: Plot Key Points for Both Functions

For \( f(x) = \log(x) \), plot points such as (1,0), (10,1), (0.1,-1). For \( g(x) = 10^{x} \), plot points like (0,1), (1,10), (-1,0.1). These points help visualize the functions' shapes.

Step 5: Sketch the Graph of \( f(x) = \log(x) \)

Draw the logarithmic curve starting from just right of \( x = 0 \) and increasing logarithmically; it should pass through the key points like (1,0) and (10,1). The curve will never touch the \( y \)-axis since it is undefined for \( x \leq 0 \).

Step 6: Sketch the Graph of \( g(x) = 10^{x} \)

Draw the exponential curve starting from a low point close to the \( x \)-axis when \( x < 0 \) and rapidly increasing as \( x \) becomes positive. It should pass through key points like (1,10) and resemble a steep upward slope.

Step 7: Analyze Intersection and Behavior

Notice that the curves will intersect at \( x = 1 \) because both functions equal 1 at this point. Observe how \( f(x) = \log(x) \) grows slower than \( g(x) = 10^{x} \) as \( x \) increases.

Key Concepts

Logarithmic Function

The logarithmic function, specifically the function \( f(x) = \log(x) \), represents the logarithm of \( x \) with base 10, often referred to as the common logarithm. This function has some interesting properties and characteristics.

• The function is defined only for positive values of \( x \) (i.e., \( x > 0 \)).
• The graph passes through the point \((1,0)\) because \( \log(1) = 0 \). This is because, in logarithms, any number raised to the power of 0 yields 1.

The curve of the logarithmic function approaches negative infinity as \( x \) approaches zero from the right. However, it never actually reaches the \( y \)-axis, meaning it has a vertical asymptote at \( x = 0 \). This can be visualized as a curve that starts just to the right of the \( y \)-axis and slowly rises, becoming less steep as \( x \) increases.

Exponential Function

An exponential function, like \( g(x) = 10^{x} \), is a function where a constant base is raised to a variable exponent. Here, the base is 10, and this function shows rapid growth as \( x \) increases.

• The function is defined for all real numbers, \( x \), unlike the logarithmic function which is only defined for positive numbers.
• It passes through the point \((0,1)\) since \( 10^0 = 1 \), illustrating that any number to the power of zero is 1.

As \( x \) grows larger, the value of \( g(x) \) increases spectacularly. This is because exponential growth is characterized by the constant doubling or more-exponential increase relative to the size itself. The graph of \( g(x) = 10^{x} \) thus resembles a steep, upward-sloping curve, starting near the \( x \)-axis for negative \( x \) and rapidly rising for positive \( x \).

Intersection of Functions

The intersection of functions occurs at a point where two graphs meet, meaning they have the same \( x \) and \( y \) values. In the problem above, the functions \( f(x) = \log(x) \) and \( g(x) = 10^{x} \) both intersect at \( x = 1 \).

• At this point, both functions are equal to 1, or more formally, \( f(1) = g(1) = 1 \).

This intersection indicates that, for the given set of functions, the exponential and logarithmic values coincide only at this point within the domain considered (\( x > 0 \) for \( \log(x) \)). This intersection can be helpful for comparing the growth rates and other properties of logarithmic and exponential functions.

Function Behavior Analysis

Understanding how functions behave is crucial for analyzing their graphs and intersections. Analyzing behavior involves looking at values as \( x \) approaches positive or negative extremes, as well as identifying special features such as intercepts and asymptotes.For \( f(x) = \log(x) \):

• The function is undefined for \( x \leq 0 \).
• It has a vertical asymptote at \( x = 0 \).
• The function increases, but at a decreasing rate as \( x \) grows.
• Intersects the x-axis at \((1,0)\).

For \( g(x) = 10^{x} \):

• Defined for all real values of \( x \).
• Crosses the \( y \)-axis at \((0,1)\).
• As \( x \) becomes positive, \( g(x) \) increases rapidly.
• For negative \( x \), the function approaches the \( x \)-axis but never touches it.

By combining this knowledge, one can better understand where these functions rise, fall, or remain constant, aiding in graph sketching and further mathematical analysis.