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 f and g | Features of g only |
| --- | --- | --- |
| | | |
domain of
range of
positive over the
negative over the
asymptote of
[tex]$x$[/tex]-intercept of
[tex]$( 0, \infty )$[/tex]
[tex]$(-\infty, \infty)$[/tex]
interval [tex]$(0,1)$[/tex]
interval [tex]$(0,1)$[/tex]
[tex]$x=0$[/tex]
[tex]$(1,0)$[/tex]
increasing as [tex]$x$[/tex] increases
decreasing as [tex]$x$[/tex] increases
Answer (1)
The domain of both functions is ( 0 , ∞ ) , the range is ( − ∞ , ∞ ) , the x -intercept is ( 1 , 0 ) , and the asymptote is x = 0 .
f ( x ) is increasing, positive over ( 1 , ∞ ) , and negative over ( 0 , 1 ) .
g ( x ) is decreasing, positive over ( 0 , 1 ) , and negative over ( 1 , ∞ ) .
The key features are categorized based on whether they apply to f ( x ) only, g ( x ) only, or both. The final categorization is provided in the solution. The answer is the categorization of the features of the two functions. The features are categorized in the steps above.
Explanation
Understanding the Problem We are asked to analyze the functions f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x and identify their key features. These features include the domain, range, intervals where the function is positive or negative, asymptotes, x -intercept, and whether the function is increasing or decreasing. We will then categorize these features into 'Features of f only', 'Features of both f and g', and 'Features of g only'.
Determining the Domain The domain of a logarithmic function lo g b x is all positive real numbers, i.e., 0"> x > 0 . Therefore, the domain of both f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x is ( 0 , ∞ ) .
Determining the Range The range of a logarithmic function lo g b x is all real numbers, i.e., ( − ∞ , ∞ ) . Therefore, the range of both f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x is ( − ∞ , ∞ ) .
Finding the x-intercept The x -intercept of a logarithmic function lo g b x occurs when the function equals zero. That is, lo g b x = 0 , which implies x = 1 . Therefore, the x -intercept of both f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x is ( 1 , 0 ) .
Intervals where f(x) is Positive and Negative To determine where f ( x ) = lo g 4 x is positive, we need to find when 0"> lo g 4 x > 0 . This occurs when 1"> x > 1 , so f ( x ) is positive over the interval ( 1 , ∞ ) . To determine where f ( x ) = lo g 4 x is negative, we need to find when lo g 4 x < 0 . This occurs when 0 < x < 1 , so f ( x ) is negative over the interval ( 0 , 1 ) .
Intervals where g(x) is Positive and Negative To determine where g ( x ) = lo g 0.25 x is positive, we need to find when 0"> lo g 0.25 x > 0 . Since the base 0.25 is between 0 and 1, the logarithm is positive when 0 < x < 1 . So g ( x ) is positive over the interval ( 0 , 1 ) . To determine where g ( x ) = lo g 0.25 x is negative, we need to find when lo g 0.25 x < 0 . This occurs when 1"> x > 1 , so g ( x ) is negative over the interval ( 1 , ∞ ) .
Identifying the Asymptote The vertical asymptote of a logarithmic function lo g b x is x = 0 . Therefore, the asymptote of both f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x is x = 0 .
Increasing or Decreasing: f(x) Since the base of f ( x ) = lo g 4 x is 1"> 4 > 1 , the function is increasing as x increases.
Increasing or Decreasing: g(x) Since the base of g ( x ) = lo g 0.25 x is 0.25 < 1 , the function is decreasing as x increases.
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 both f and g:
domain of ( 0 , ∞ )
range of ( − ∞ , ∞ )
x -intercept of ( 1 , 0 )
asymptote of x = 0
Features of g only:
decreasing as x increases
positive over the interval ( 0 , 1 )
negative over the interval ( 1 , ∞ )
Final Answer The key features of the functions f ( x ) = lo g 4 x and g ( x ) = lo g 0.25 x have been identified and categorized. The features are:
Features of f only:
increasing as x increases
positive over the interval ( 1 , ∞ )
negative over the interval ( 0 , 1 )
Features of both f and g:
domain of ( 0 , ∞ )
range of ( − ∞ , ∞ )
x -intercept of ( 1 , 0 )
asymptote of x = 0
Features of g only:
decreasing as x increases
positive over the interval ( 0 , 1 )
negative over the interval ( 1 , ∞ )
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, determining the loudness of sound in decibels, and modeling population growth or radioactive decay. Understanding the properties of logarithmic functions, such as their domain, range, intercepts, and asymptotes, is crucial for interpreting and analyzing data in these fields. For example, in seismology, the Richter scale uses logarithms to quantify the magnitude of an earthquake. An earthquake of magnitude 6 is ten times stronger than an earthquake of magnitude 5. Similarly, in acoustics, the decibel scale uses logarithms to measure sound intensity. A sound of 60 decibels is twice as loud as a sound of 50 decibels. These applications demonstrate the practical importance of understanding logarithmic functions.
