Question

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

\(y = {\bf{lo}}{{\bf{g}}_{{\bf{1}}.{\bf{5}}}}x\),\(y = {\bf{ln}}x\), \(y = {\bf{lo}}{{\bf{g}}_{{\bf{10}}}}x\), \(y = {\bf{lo}}{{\bf{g}}_{{\bf{50}}}}x\)

Short answer

The graph is shown below:

The graphs are related as:

(i) The graph of all the functions is increasing.

(ii) All functions are approaching to \(\infty \) as \(x \to \infty \).

iii) The rate of increase of functions with larger base is lower.

Step-by-step solution

Simplify the function using logarithmic properties

The equation \(y = {\log _{1.5}}x\) can be written as shown below:

\(\begin{array}y = {\log _{1.5}}x\\ = \frac{{\ln x}}{{\ln 1.5}}\end{array}\)

The equation \(y = {\log _{50}}x\) can be written as shown below:

\(y = \frac{{\ln x}}{{\ln 50}}\)

Sketch the graph of given functions

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

1. In the graphing calculator, select “STAT PLOT” and enter the equations \({\log _{1.5}}x\), \(\ln x\), \({\log _{10}}x\), and \({\log _{50}}x\).
2. Enter the graph button in the graphing calculator.

The figure below represents the graph of the functions \(y = {\log _{1.5}}x\), \(y = \ln x\), \(y = {\log _{10}}x\), and \(y = {\log _{50}}x\).

Find the relation between the graphs

Following are the observations from the graphs:

(i) The graph of all the functions is increasing.

(ii) All functions are approaching to \(\infty \) as \(x \to \infty \).

(iii) The rate of increase of functions with a larger base is lower.