Problem 115 - Algebra and Trigonometry

Question

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Step-by-Step Solution

Answer

In the interval between 0 and 1, the graph with the lower base will be on top but in the interval between 1 and ∞, the graph with the higher base will be on top. This behavior of graphs is common for the function \(y=log_{b}x\)

Step 1: Graphing

Use the change-of-base property and a graphing utility to graph the three functions \(y=log_{3}x, y=log_{25}x, y=log_{100}x\) in the same viewing rectangle.

Step 2: Analysing the Interval (0,1)

Now analyze the graphs in the interval (0,1). Here, the graph of \(y=log_{3}x\) will be on top, followed by the graph of \(y=log_{25}x\), and the graph of \(y=log_{100}x\) will be at the bottom. The function gets lower as the base increases.

Step 3: Analysing the interval (1, ∞)

Next, analyze the graphs in the interval (1, ∞). In this range, the graph of \(y=log_{100}x\) will be on top, followed by the graph of \(y=log_{25}x\), and the graph of \(y=log_{3}x\) will be at the bottom. In this interval, the function gets higher as the base increases.

Step 4: Generalization

For the function \(y=log_{b}x\) where b > 1, the graph will show that when x belongs to the interval between 0 and 1, lower base graph will be on top. However, for x in the interval between 1 to infinity, the graph of higher base will be on top. This is because the function gets lower as the base increases within the range of (0,1) and gets higher as the base increases within the range of (1, ∞)