Exponential & Logarithmic Functions
Exponential Functions:

Definition:     The function f is an exponential function if

     f(x) = b x

where b is a positive constant other than 1 and x is any real number. The number x is called exponent and b is called the base.

Notice that domain and range are given by: Dom(f) = & Range(f) = (0,+\infty)

Example 1.    Sketch the graph of f if f(x)=2x.

Solution.     Coordinates of some points are listed in the following table:

x

-4

-3

-2

-1

0

1

2

3

4

2x

1/16

1/8

1/4

1/2

1

2

4

8

16

If you want to see the points being plotted one by one and finally the graph, erase the graph and click on each xi above to see point P(xi, yi) being plotted. Finally, click on 2x to see the graph.

This applet graphs a fixed function 2x and the function A0x which changes as you change the value of A0 .

Logarithmic Functions:

Definition:     The function f defined by

     f(x) = log a x

for all positive real numbers x is called the logarithmic function with base a

Here Dom(f) = (0,+\infty) and Range(f) =

Example 2.    Sketch the graph of f if f(x)=log2 x.

Solution.     Coordinates of some points are listed in the following table:

x

0.1

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

log2 x

-3.322

0.000

1.000

1.585

2.000

2.322

2.585

2.807

3.000 

Next, see the above points ploted and a sketch of the graph:

If you want to see the points being plotted one by one and finally the graph, erase the graph and click on each xi above to see point P(xi, yi) being plotted. Finally, click on log2x to see the graph.

 

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© 1998 Teresinha Kawasaki