In the problems below, [tex]f(x) = log_2x[/tex] and [tex]g(x) = log_{10}x[/tex]. How are the graphs of [tex]f[/tex] and [tex]g[/tex] similar? Check all that apply.
A. Both have a y-intercept of 1.
B. Both increase from left to right.
C. Both have an asymptote of [tex]x = 0[/tex].
D. Both have a domain of all real numbers.
Answer (2)
Logarithmic functions do not have a y-intercept.
Logarithmic functions with a base greater than 1 increase from left to right.
Logarithmic functions have a vertical asymptote at x = 0 .
The domain of logarithmic functions is 0"> x > 0 .
The correct options are:
Both increase from left to right.
Both have an asymptote of x = 0 .
Therefore, the answer is: Both increase from left to right, Both have an asymptote of x = 0
Explanation
Understanding the Problem We are given two functions, f ( x ) = lo g 2 x and g ( x ) = lo g 10 x , and we need to identify the similarities between their graphs from the given options.
Analyzing the Options Let's analyze each option:
Both have a y-intercept of 1: A y-intercept occurs where x = 0 . However, logarithmic functions are not defined at x = 0 . Therefore, they do not have a y-intercept. This statement is false.
Both increase from left to right: Logarithmic functions, y = lo g b x , increase from left to right when the base 1"> b > 1 . Since both functions have bases greater than 1 ( 2 and 10 ), they both increase from left to right. This statement is true.
Both have an asymptote of x = 0 : Logarithmic functions have a vertical asymptote at x = 0 . This is because as x approaches 0 , the value of the logarithm approaches negative infinity. This statement is true.
Both have a domain of all real numbers: The domain of a logarithmic function y = lo g b x is 0"> x > 0 . Therefore, the domain is not all real numbers. This statement is false.
Conclusion Based on the analysis, the correct statements are:
Both increase from left to right.
Both have an asymptote of x = 0 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes (Richter scale), the loudness of sound (decibels), and the acidity of a solution (pH scale). Understanding the properties of logarithmic functions, such as their increasing behavior and asymptotes, helps us interpret and analyze these phenomena. For example, the Richter scale uses a base-10 logarithm to quantify the magnitude of an earthquake. An increase of 1 on the Richter scale corresponds to a tenfold increase in the amplitude of the seismic waves.
The graphs of both functions f ( x ) = lo g 2 x and g ( x ) = lo g 10 x increase from left to right and have an asymptote at x = 0 . However, they do not have a y-intercept nor do they have a domain of all real numbers. Therefore, the correct choices are: Both increase from left to right and both have an asymptote of x = 0 .
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