1. Exponential Functions and an Introduction to Logarithmic Functions
On completion of this module you will be able to:
recognise the general form of the exponential and logarithmic functions
recall important features of the exponential and logarithmic functions
draw the graphs of exponential and logarithmic functions
create simple models using exponential functions (compound interest and population growth)
understand the logarithmic function and its relationship to the exponential function

2. Exponential Functions
Exponential functions have the form
where b > 0 and b  (Why?)
If b = 1 then 1x=1 for all values of x and the graph is a horizontal line through y = 1.
If b is less than zero then x = ½ produces problems e.g. at we have which cannot be evaluated.

3. Notes:
Do not confuse bx with xb. The first is an exponential function; the second is a power function.
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
All exponential functions of this form pass through the point (0, 1) since b0 = 1 for any value of b.

4. (never quite horizontal or vertical)
f(x)
(0,1) x
(never quite horizontal or vertical)

5. A more general definition describes exponential functions by
The earlier model had a = 1.
All exponential functions of this form pass through the point (0, a) since b0 = 1 for any value of b.

6. (never quite horizontal or vertical)
f(x)
(0,a) x
(never quite horizontal or vertical)

7. Sketching Exponential Curves
Remember that the coordinate (0, 1) or (0, a) is a point on the curve.
Select a further five to six x values and calculate the value of the function at these points.
Usually pick x values near the origin since the value of the function may become very, very large or very, very small, very quickly.

8. Evaluate the function at your selected x values.
Often produce a table of the x and function values as in the following example.

9. Example: Curve sketching
Sketch the function
x f(x)
Because of scaling it is only practical to plot about 3 of these points

10. Additional points:
x f(x)

12. Modelling with exponential functions
Compound Interest
This equation describes the process of having interest earned being invested again to earn more interest etc.
S is the compound amount, or the amount that the principal, P, will compound to.
S is a function of n, the number of interest periods.
The r is the periodic interest rate.

13. Interest is usually given as a nominal or annual percentage rate.
The periodic rate, which we used in the previous formula, is the annual rate divided by the number of interest periods per year.

14. is an example of the exponential formula

15. Example
What will $2500 invested at 12.5% paid quarterly for 3 years amount to?

16. Example
Over three years, $2500 will amount to $

17. Population Growth
A general equation describing population growth is given by
where

18. Population Growth
e is called the natural exponential function (mentioned in Week 1).
It is an irrational number like .
It occurs as a natural consequence of the calculus.

19. Example: Forecasting Population
The growth of a particular population is described by
Currently the population is Graph the population over the next 10 years. What is the year forecast?

20. Example: Forecasting Population
At t = 0,
At t = 2,
At t = 4,
etc

21. t Pf
3700 2
4519 4
5520 5
6100 6
6742 8
8235 10
10058

22. The population can be expected to grow to 6100 in 5 years.

23. Logarithmic Functions
The logarithmic function is the inverse of the exponential function.
Consider the function
f(x)=10 x
This is an exponential function with a base of 10.

24. Logarithmic Functions
f(x)=10 x
We are familiar with substituting values of x into functions, so for example when x = 2, f(x) = 100.
But what if we want to ask the question, what is x when f(x) = 100? What about when f(x) = 50?
Problems like this can be solved using logarithms.

25. The definition of the logarithm is:
So given an exponential function of the form writing means exactly the same thing.
Logarithms are another way of finding the value of the exponent, y.

26. Example: Applying the definition of logs
Solve for x:

27. Solution:
Using the definition, we can rewrite this as
and so

28. Solution:
So the value of the unknown must be

29. Most calculators have two buttons for log functions: log and ln.
The log button
log is inverse of 10x (inverse of the exponential function with base 10).
log means log10.

30. The ln button
ln x is the inverse of ex (inverse of the natural exponential function).
ln stands for natural logarithm and is a shorthand way of writing loge.

31. Remember:
and

32. Evaluating some log and ln values
Check the following on your calculator:
Note that you will get an error message if you try to take the log or ln of any value less than or equal to zero.

33. Example: Graph of f(x) = log x
First produce a table of 5 to 6 function values.

35. All log functions of the form logbx pass through (1,0).
Compare this with exponential functions of the form bx which always pass through (0,1).
This is because they are inverses and so reflections of each other in the line y = x.

36. Solving Logarithmic Equations
We will now use the definition of logs to solve some more equations.
Example

37. Answer

39. But, substituting this into the original equation gives
which is undefined, so the answer is “No solution”.

40. So there are no real solutions.

41. Since we know that 92=81, then x+3 must be equal to 2 and so

42. Since we know that 33=27, then 2x+3 must be equal to 3 and so