Module 15: EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Exploring Calculus
In this module we will look at two functions which are very different from those which we have discussed before--the exponential function and the logarithmic function . They are two of the most useful functions for solving real world problems in biology, business, chemistry, engineering and physics.
Critical Thinking Questions
1. a. In each of the functions f (x) = 
, g (x) = 
, and h (x) = 
, is the base a variable or a constant?
b.
Display the graph for the three functions above and fill in the chart below for for x = -5, -2, 0, 2, 5 ?
|
x |
f (x) |
g (x) |
h (x) |
|
-5 |
. |
. |
. |
|
-2 |
. |
. |
. |
|
0 |
. |
. |
. |
|
2 |
. |
. |
. |
|
5 |
. |
. |
. |
c. For x > 0 , which function grows the fastest and which function decays the fastest?
2. The function y = b x, b > 0 , (if b equals 1, it is not an exponential function. Why ? ) is called the exponential function. It behaves like y = 
, when b >_____ , and like y = 
when 0 < b < _____.
3. The most important base is the number e .
a. Using your calculator complete the following table.
|
n |
|
|
|
1 |
. |
. |
|
10 |
. |
. |
|
100 |
. |
. |
|
1,000 |
. |
. |
|
10,000 |
. |
. |
|
100,000 |
. |
. |
|
1,000,000 |
. |
. |
|
10,000,000 |
. |
b. Now use your calculator to find the approximate value for e to nine decimal places.
c. Fill in the blanks.
_____
=_____ .
d. Sketch a graph of y = 
, y = 
and y = 
for the x-values - 2, - 1 , 0 , 1 and 2 on the grid below.
The inverse of the function y = f (x) is the function x = f (y) , which when solved for y is written as 
. Read 
as: f inverse of x .
When the point (x , y) is on the graph of y = f (x) , its inverse point (y , x) is on the graph of .
4. Given y = 
a. Switch x and y and solve for y. The resulting function y = _____ is the inverse of y = 
.
b. If we know that the point (2, 8) is on the graph of f (x) = 
, then we know the point _____ is on the graph of y = f -1(x).
Display the graph.
c. What is the equation of the line graphed in yellow? What relationship do the graphs of f (x) and f -1(x) have to each other and to this line?
The domain of is the range of y = f (x), and the range of is the domain of y = f (x).
5. What is the inverse of y = 
? State its domain and range. 
Display the graph. Hint
6. a. What is the relationship between the graphs of 
(the natural exponential function) and 
(the natural logarithmic function)? (This may--but is generally not--be written as 
.)
Display the graph.
b. What are domain and range of each function?
Skill Exercises
Review the rules of exponents and logarithms and take the practice Test.
1. a. Any exponential equation may be written as a logarithmic equation. That is, the equation y = 
may be written as_____ =
_____ .
b. In particular, since 
= _____ , we may also write 
_____ = _____ ; and since 
= _____ , we may also write 
_____ = _____ .
c. Furthermore, since 
= _____ , we can also say that ln _____ = 0 ; and since 
= _____ , we can also say that ln _____ = 1 .
2. Use your calculator to find ln 2 , ln 3 , and ln 4 .
Display the graph. Note the area under the function y = 
above the t-axis on each of the intervals [1,2] , [1,3] , and [1,4] .
a. Use the above information to help you fill in the blank in the upper limit of the integral below.
b. If F(x) = 
, what is F ' (x)? (This is the Fundamental Theorem of Calculus.)
c. Now, if F(x) is the integral in part a. above (with the upper limit correctly identified), what is F ' (x)?
3. Let y = 
.
Take ln of both sides of this equation : ln y = _____ .
Use logarithm rules and simplify : ln y = _____ .
Implicitly differentiate this equation : _____ = _____.
Hint
Solve for 
: 
= _____ .
Now, rewrite "y" in terms of x : 
= _____ .
Explain what you have derived.
4. Suppose that 
is proportional to y , written 
α y .
Then we know that 
= _____ y .
We may then separate the variables to obtain 
= k_____ .
Now, integrate both sides of the equation to obtain _____ y = _____. 
Hint
Solve the above equation for "y" , by letting y(0) = y0 , to obtain y = _____.
Rewrite the equation above as y = _____e k t. The value of y at t = 0 is called y0.
Rewrite the above equation using y0 .
You have derived the Law of Growth and Decay , y = _____.
5. Use the law of growth and decay to solve the following problem. The Polonium isotope 210 Po has a half-life of approximately 140 days. If a sample weighs 20 milligrams initially, how much remains after t days? Approximately how much will be left after two weeks?
Hint
.
This document was served by the faculty/staff web server and is not an official college document.
This document was last modified on 19-Feb-01.
This document was served by the faculty/staff web server and is not an official college document.
This document was last modified on 19-Feb-01.
This document was served by the faculty/staff web server and is not an official college document.
This document was last modified on 21-Feb-01.
This document was served by the faculty/staff web server and is not an official college document.
This document was last modified on 16-Jul-01.