Question

49-50 Use formula 11 to graph the given functions on a common screen. How are these graphs related?

\(y = {\bf{ln}}x\), \(y = {\bf{lo}}{{\bf{g}}_{\bf{8}}}x\), \(y = {e^x}\), \(y = {{\bf{8}}^x}\)

Short answer

The graph is shown below:

The graphs are related as:

i) The rate of increase of of graph of \({8^x}\) is the highest.

ii) The graph of \({\log _8}x\) is approaching \(\infty \) more slowly than the graph of \(\ln x\).

Step-by-step solution

Sketch the graph of given functions

The graph of \(\ln x\) is the reflection of \({e^x}\) about the line \(y = x\). Similarly, the graph of \({8^x}\) is reflection of \({\log _8}x\) about \(y = x\).

Use the following steps to plot the graph of given functions:

1) In the graphing calculator select “STAT PLOT” and enter the equations \({8^x}\), \({e^x}\), \(\ln x\), and \({\log _8}x\).

2) Enter the graph button in the graphing calculator.

The figure below represents the graph of the functions \(y = \ln x\), \(y = {\log _8}x\), \(y = {e^x}\), and \(y = {8^x}\).

Find the relation between the graphs

Following are the observations of the graphs:

i) The rate of increase of of graph of \({8^x}\) is the highest.

ii) The graph of \({\log _8}x\) is approaching \(\infty \) more slowly than the graph of \(\ln x\).