Use the Graphing tool to graph the functions [tex]f(x)=\log _4 x[/tex] and [tex]g(x)=\log _{0.25} x[/tex]. Then identify the key features of each graph.
Drag each feature to the correct location on the table.
| Features of f only | Features of both [tex]f[/tex] and [tex]g[/tex] | Features of [tex]g[/tex] only |
| --- | --- | --- |
| | | |
domain of [tex](0, \infty)[/tex]
range of
[tex](-\infty, \infty)[/tex]
positive over the interval [tex](0,1)[/tex]
negative over the interval [tex](0,1)[/tex]
asymptote of [tex]x = 0[/tex]
[tex]x[/tex]-intercept of [tex](1,0)[/tex]
increasing as [tex]x[/tex] increases
decreasing as [tex]x[/tex] increases
Answer (1)
f ( x ) = lo g 4 x is an increasing function with domain ( 0 , ∞ ) , range ( − ∞ , ∞ ) , x-intercept ( 1 , 0 ) , and asymptote x = 0 .
g ( x ) = lo g 0.25 x is a decreasing function with domain ( 0 , ∞ ) , range ( − ∞ , ∞ ) , x-intercept ( 1 , 0 ) , and asymptote x = 0 .
f ( x ) is positive over ( 1 , ∞ ) and negative over ( 0 , 1 ) , while g ( x ) is positive over ( 0 , 1 ) and negative over ( 1 , ∞ ) .
The common features are domain ( 0 , ∞ ) , range ( − ∞ , ∞ ) , x-intercept ( 1 , 0 ) , and asymptote x = 0 . The unique features are the increasing/decreasing nature and the intervals where they are positive/negative. $\boxed{{\text{See the table in the solution}}}.
Explanation
Understanding the Problem We are given two logarithmic functions, f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x , and we need to identify their key features and categorize them. The features to consider are the domain, range, intervals where the function is positive or negative, asymptotes, x-intercept, and whether the function is increasing or decreasing.
Analyzing f(x) Let's analyze the function f ( x ) = lo g 4 x . Since the base 4 is greater than 1, this is an increasing logarithmic function. The domain of a logarithmic function is all positive real numbers, so the domain is ( 0 , ∞ ) . The range is all real numbers, ( − ∞ , ∞ ) . The x-intercept occurs when f ( x ) = 0 , which means lo g 4 x = 0 . This happens when x = 4 0 = 1 , so the x-intercept is ( 1 , 0 ) . The function is positive when 1"> x > 1 and negative when 0 < x < 1 . The vertical asymptote is x = 0 .
Analyzing g(x) Now let's analyze the function g ( x ) = lo g 0.25 x . Since the base 0.25 is between 0 and 1, this is a decreasing logarithmic function. The domain of a logarithmic function is all positive real numbers, so the domain is ( 0 , ∞ ) . The range is all real numbers, ( − ∞ , ∞ ) . The x-intercept occurs when g ( x ) = 0 , which means lo g 0.25 x = 0 . This happens when x = ( 0.25 ) 0 = 1 , so the x-intercept is ( 1 , 0 ) . The function is positive when 0 < x < 1 and negative when 1"> x > 1 . The vertical asymptote is x = 0 .
Categorizing the Features Now, let's categorize the features:
Features of f only:
increasing as x increases
positive over the interval ( 1 , ∞ )
negative over the interval ( 0 , 1 )
Features of g only:
decreasing as x increases
positive over the interval ( 0 , 1 )
negative over the interval ( 1 , ∞ )
Features of both f and g:
domain of ( 0 , ∞ )
range of ( − ∞ , ∞ )
asymptote of x = 0
x-intercept of ( 1 , 0 )
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth or decay in biology. Understanding the properties of logarithmic functions, such as their domain, range, and asymptotes, is crucial for interpreting and analyzing data in these fields. For example, the Richter scale uses a base-10 logarithm to quantify the magnitude of an earthquake. An earthquake with a magnitude of 6 is ten times stronger than an earthquake with a magnitude of 5. Similarly, the pH scale uses a base-10 logarithm to measure the acidity or alkalinity of a solution. A solution with a pH of 3 is ten times more acidic than a solution with a pH of 4.
