Chapter 3.2, Problem 34E

To determine

To plot: the functions $f\left(x\right)=5^x$ and $g\left(x\right)={\log }_{5}x$ on the same coordinates.

Explanation of Solution

Given:

The functions are $f\left(x\right)=5^x$ and $g\left(x\right)={\log }_{5}x$ .

Calculation:

Consider $y=5^x$ and $y={\log }_{5}x$ ,

Take some of values of x and find the values of y in first equation as follows:

x	y
0	1
-2	0.04
0.2	1.38
0.1	1.175
0.3	1.621
0.5	2.236

Draw above coordinates and join them to make plot of $y=5^x$ as shown below:

  

Now, consider second equation $y={\log }_{5}x$ and simplify as follows:

  $x=5^y$

From above equation, it is clear that both the functions have opposite axis means y replaces x and x replaces y.

Thus, the coordinates are just same by changing coordinates x and y as given below:

y	x
0	1
-2	0.04
0.2	1.38
0.1	1.175
0.3	1.621
0.5	2.236

Draw above coordinates and join them to make plot of $x=5^y\text{}\text{or}y={\log }_{5}x$ as shown below:

  

Take both the plots on the same plane as shown below: