Which is the graph of $y=5 \log (x+3)$?
Answer (1)
The function is y = 5 lo g ( x + 3 ) .
The vertical asymptote is x = − 3 .
The x-intercept is ( − 2 , 0 ) .
The graph passes through ( 7 , 5 ) and is increasing.
Explanation
Understanding the Function We are asked to identify the graph of the function y = 5 lo g ( x + 3 ) . This is a logarithmic function with base 10. We need to consider the transformations applied to the basic logarithmic function and identify key features such as the vertical asymptote and intercepts.
Finding the Vertical Asymptote The basic logarithmic function y = lo g ( x ) has a vertical asymptote at x = 0 . The function y = lo g ( x + 3 ) is a horizontal translation of the basic logarithmic function by 3 units to the left. Therefore, the vertical asymptote is at x = − 3 .
Finding the x-intercept To find the x-intercept, we set y = 0 and solve for x : 0 = 5 lo g ( x + 3 ) lo g ( x + 3 ) = 0 x + 3 = 1 0 0 = 1 x = 1 − 3 = − 2 So the x-intercept is at ( − 2 , 0 ) .
Finding Another Point To find another point on the graph, we can choose a convenient value for x . Let's choose x = 7 . Then: y = 5 lo g ( 7 + 3 ) = 5 lo g ( 10 ) = 5 ( 1 ) = 5 So the point ( 7 , 5 ) is on the graph.
Identifying the Graph Since the coefficient of the logarithm is positive (5), the graph is increasing. The graph approaches the vertical asymptote x = − 3 from the right. Based on the vertical asymptote at x = − 3 , the x-intercept at ( − 2 , 0 ) , the point ( 7 , 5 ) , and the increasing nature of the graph, we can identify 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 determining the pH of a solution in chemistry. Understanding how to graph and interpret logarithmic functions is essential for analyzing and predicting behavior in these scenarios. For example, if we know the intensity of an earthquake, we can use the Richter scale, which is a logarithmic scale, to determine its magnitude. Similarly, in finance, logarithmic scales are used to analyze stock market trends and investment growth.
