Select the correct answer.
Which graph is the graph of [tex]$f(x)=3 \log _2(x+1)$[/tex]?
A.
B.
Answer (1)
The function f ( x ) = 3 lo g 2 ( x + 1 ) has a vertical asymptote at x = − 1 .
The x-intercept is at ( 0 , 0 ) .
The function is increasing as x increases.
Graph A matches these characteristics, so the answer is A .
Explanation
Analyze the function We are given the function f ( x ) = 3 lo g 2 ( x + 1 ) and asked to identify its graph from two options. Let's analyze the key features of this function to determine the correct graph.
Find the vertical asymptote First, note that the argument of the logarithm must be positive, so 0"> x + 1 > 0 , which means -1"> x > − 1 . This tells us that the function has a vertical asymptote at x = − 1 .
Find the x-intercept Next, let's find the x-intercept by setting f ( x ) = 0 : 3 lo g 2 ( x + 1 ) = 0 lo g 2 ( x + 1 ) = 0 x + 1 = 2 0 x + 1 = 1 x = 0 So the x-intercept is at ( 0 , 0 ) .
Analyze the behavior of the function Since the base of the logarithm is 2, which is greater than 1, the function is increasing as x increases. The vertical stretch by a factor of 3 does not change the location of the vertical asymptote or the x-intercept, but it makes the graph increase more rapidly.
Compare the graphs Now, let's examine the two graphs. Graph A has a vertical asymptote at x = − 1 and passes through the point ( 0 , 0 ) . It also increases as x increases. Graph B has a vertical asymptote at x = 1 and does not pass through the point ( 0 , 0 ) . Therefore, Graph A is the correct graph.
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and calculating the pH of a solution. Understanding the properties of logarithmic functions, such as the vertical asymptote and x-intercept, is essential for interpreting and analyzing these applications. For example, in seismology, 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. By understanding the logarithmic relationship, we can better comprehend the relative sizes and impacts of different earthquakes.
