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Magazine Chapter 6: Problem 36
For the following exercises, sketch the graphs of each pair of functions on the same axis.$$f(x)=\log (x) \text { and } g(x)=10^{x}$$
Short Answer
Expert verified
Graph \( f(x) = \log(x) \) starting just right of \( x=0 \); graph \( g(x) = 10^{x} \) from a low point upward.
Step by step solution
01
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.
02
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.
03
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.
04
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.
05
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 \( x \leq 0 \).
06
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.
07
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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, \( f(1) = g(1) = 1 \).
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 \((1,0)\).
- 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.
