Problem 115 - College Algebra Essentials

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

a. In the interval (0,1), the graph of \(y=\log _{100} x\) is the highest and \(y=\log _{3} x\) is the lowest. b. In the interval (1, \infty), \(y=\log _{3} x\) is the highest graph and \(y=\log _{100} x\) is the lowest. c. In general, for \(y = \log_{b}x\) where \(b > 1\), the graph with a smaller base is higher for x > 1, and the graph with a larger base is higher for x < 1.

Step 1: Graph the Functions

Start with graphing the three functions \(y=\log _{3} x\), \(y=\log _{25} x\), and \(y=\log _{100} x\) on the same graph. This can be done using a graphing utility.

Step 2: Analyze the Graphs in the Interval (0,1)

Observe the graphs in the interval (0,1). Determine which graph is at the top or bottom in this interval.

Step 3: Analyze the Graphs in the Interval (1, \infty)

Next, study the behaviour of the graphs in the interval (1, \infty). Again, determine which graph is at the top or bottom in this interval.

Step 4: Formulate a General Rule

Based on the observations in the previous steps, formulate a general rule about which function has the highest and lowest graph in each interval for \(y = \log_{b}x\), where \(b > 1\).