Problem 115 - College Algebra

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 \( (0,1) \), \( y=\log _{100} x \) is at the top and \( y=\log _{3} x \) is at the bottom. In the interval \( (1, \infty) \), \( y=\log _{3} x \) is at the top and \( y=\log _{100} x \) is at the bottom. In general, for \( y=\log _{b} x \) , if b is larger, the graph will be beneath in the interval \( (1, \infty) \), and vice versa.

Step 1: Graphing the functions

First thing to do is graphing the functions \(y=\log _{3} x,\) \(y=\log _{25} x\) and \(y=\log _{100} x\). Important point here is setting the same viewing rectangle for each function. After graphing, three different curves for each function should be visible.

Step 2: Analyzing the interval (0,1)

For interval \((0,1)\), directly inspect from the graph to determine which function is at the top and which is at the bottom. Each graph y= log_bx will yield different results based on its base value. Normally, the larger the base, the smaller the value of the logarithmic function is within this interval, meaning it will be at the bottom of the graph.

Step 3: Analyzing the interval (1, Infinity)

For interval \((1, Infinity)\), again directly inspect the graph to determine. However, contrasts to the previous interval, the smaller the base, the smaller the value of the logarithmic function within this interval, meaning it will be at the bottom of the graph.

Step 4: Generalization

To generalize, if \(y=\log _{b} x\), for interval \((0,1)\), the function with the larger base value will be at the bottom. On the other hand, for interval \((1, Infinity)\), the function with the smaller base value will be at the bottom. This is because the larger the base, the slower the logarithmic growth, making its graph lower than the others in the positive field, and higher in the negative field.