Exponential and Logarithmic Functions
Instructional Notes
Math B30
Exponential and Logarithmic Functions
Instructional Notes
Students could be presented with some real-world examples, such as chain letters, the checkerboard problem, or interest calculations (see next column), to show how exponents are encountered in various situations. These all represent exponential functions. From these examples, the definition of an exponential function could be introduced. A general form is f(x) = ax, where a > 0, and x 
R.
The logarithmic function can be introduced as the inverse of the exponential function. If y = ax is the exponential function, then the logarithmic function can be written as x =ay. This can then be written in logarithmic form as y = loga x. E.g.: loga x is the power to which we raise a to get x. Students can be given some practice converting from one form to the other.
Examples.
8 = 23 is equivalent to log2 8 = 3
c = ba is equivalent to logb c = a
2 log23 = 3
5 log5q = q
This section includes the review of the laws of exponents for integral and rational values, the extension of these laws to include irrational exponents, the solution of equations involving exponents, and the introduction of the laws of logarithms for base ten. It is important that students gain an understanding of the laws of logarithms and how they relate to the laws of exponents. The solution of equations involving exponents, by converting to the same base, is also useful in further work, especially series.
The teacher may wish to structure the review of laws of exponents as a series of exercises designed to allow students to review these on their own. The introduction of irrational exponents could be done by pairs of students working on a few exercises, and discussing their procedures with each other. (COM)
The solution of equations could be done by example, allowing students to propose possible procedures, and then checking the result, while the laws of logarithms could be introduced by the teacher as a class lesson, using didactic questioning.
The laws of logarithms should include those for:
multiplication (log mn = log m+log n),
division (log x/y = log x - log y),
powers (log xn = n log x), and
roots (log n
x = 1/n log x).
In this section, the laws of logarithms should be adapted to allow for work with bases other than 10. The exercises and examples chosen should be similar to those in the previous section. The teacher may wish to begin by having the students discuss the solutions to log statements such as log2 8 = ?, log6 (1/216) = ? , loga ab = ?
The introduction of loga n = logb n/(logb a)
as a means of changing bases, might also be introduced at this point.
Students should be given a series of graphs to sketch on a graphic calculator or computer, transcribing the general shape to paper, or obtaining print-outs for further analysis.
The first such series of graphs could be y = ax, where a>0. The second series could be y = loga x. These two families of graphs could be compared to determine some characteristics that each type has in common, and the relationship between the two types.
Other types of exponential and logarithmic functions could then be graphed, using graphic calculators or computers.
The teacher may wish students to obtain a table of values for one or two of the first graphs, in order to understand the calculations that are necessary before the graphs appear on a screen.
This section is intended to have students become familiar with exponential and logarithmic graphs that have the same properties as those in the previous section, but have constants included which cause shifts in the graphs. By choosing a different point as the origin, the shifts can be counteracted and the same properties become much more evident.
Once students understand how to shift or translate the axes, a few exercises might be given for them to graph. These exercises might be relatively simple, such as graphing y = 3 + 2x, and then shifting the axes, or they might be more difficult, involving several shifts, such as y = 2 + 2x-3. This could also be written in the form y-2 = 2x-3.
Similar types of graphs could be done for logarithmic functions.
This section could be introduced by asking students to attempt to determine how long it would take an investment to quadruple in value if it grew at an annual rate of 8%. Students could work individually, in pairs or in small groups to discuss strategies, pertinent information, how to make use of exponents or logarithms, and attempt to solve the problem. Should the students seem to be having difficulty starting, a second problem might be posed. The second problem could be stated as; "What number, when cubed, has a result of 500?" Once students are able to solve this, they may return to the first problem. (CCT)
Alternatively, the teacher may wish to do a class example of an exponential or logarithmic equation and then have students complete a few exercises.
Most of the text resources have adequate numbers of real-world problems. The teacher may wish to have the students work on a few of these in pairs and have the entire class take part in the `taking-up' of these initial exercises. Most of these problems deal with growth and decay, and require formulas in some cases. These formulas are normally part of, or immediately preceding, the problems given in the resource texts.
Adaptation of these problems to events and places closer to the student might generate more interest, but will take extra preparation time.
When students have had some successful experience with the initial exercises, they might be provided with some further problems for practice.
(IL)
Begin with a brief review of the definition of sequence and some of the terms associated with an arithmetic sequence as taken in Mathematics 10.
Once the review is complete, students could be given some examples of geometric sequences and asked to determine the next few terms of the sequence. When they have completed these, the definitions and terms associated with geometric sequences can be formalized. The basic terms are the first term, the common ratio between successive terms, and the number of terms. Various resources employ different variables to designate each, but the definitions and usage remain constant.
Student exercises for this topic can be identification exercises, completing the next few terms, or the generation of geometric sequences.
The teacher may initially instruct students to determine a specific term of a given sequence, without further instruction. Students could work individually or in pairs on a short series of exercises of this nature. The exercises could be designed so that the students can begin to identify a procedure other than continually multiplying successive terms by r.
If the class is not able to identify the general rule, the teacher can help them formalize the process by demonstrating the rule for finding the nth term of a geometric sequence through class examples.
Students should be able to demonstrate their understanding of the general rule by correctly applying it to a short series of assigned exercises where they are instructed to find the nth term, or by being able to develop the rule from the general geometric sequence a, ar, ar2, ar3, .........
Students could work individually, in pairs, or small groups, and have a problem posed to them. The information provided would allow them to determine a, and tn. They would be instructed to determine a specific number of terms (geometric means) between the two.
They would be expected to provide the solution and justify their answer. When this exercise has been completed, the entire class can work together to formalize the procedure, including specific definitions needed for this topic. A short series of exercises on calculating means could then be given as an assignment.
The definition of a geometric series and the formulas required for geometric series should be introduced. Some of the better students may wish to derive the formula on their own, given a starting point. However, for most students, the teacher could lead the class through the derivation.
The class should practise using these formulas to find the indicated sum of a few series. The summation notation for geometric series should also be introduced in this section. The students should become familiar with all terms associated with the summation notation.
The class should be given a variety of exercises that allows them to practise using these formulas and notations.
This section is placed here in order for students to take inventory of some of the terms and concepts they will need in the following sections, where applications of their knowledge will be expected. Students should be able to define and explain each of the terms listed here, as well as demonstrate their knowledge of the terms.
In this section, the specific terms of annuity and present value are the only ones not encountered previously in mathematics, and should be introduced to the class. Most of the students will have encountered these terms through their everyday life, or in other classes, and may be able to provide background explanation, and know examples of applications. (COM)
The teacher can present the class with a set of sequences which have an infinite number of terms, and have students plot the first several terms of each sequence on a graph. This should enable students to determine visually whether the sequence has a limit (asymptote). Example.
Plot each sequence on a graph.
a) 1, 1/3, 1/9, 1/27, 1/81,.......
b) 2, 3, 2, 3, 2, 3, 2, 3,............
c) 2, 4, 6, 8, 10,.........
d) 1/3, 3/5, 5/7, 7/9, 9/11.........
From this introduction, the teacher can then introduce the concept of a limit and the definitions of converging and diverging sequences. Some examples and exercises involving these concepts can be completed by the class.
Students could be given one or two exercises in which to attempt to determine the sum of a given series. These might include the following;
a) 9, 6, 4, 8/3,............. and
b) 2, 3, 4.5, 6.75, .........
Students should be able to identify a and r in both cases and substitute into one of the summation formulas. For the first, they should obtain 
and 
When they arrive at this point, they can be instructed to utilize their work with limits in the previous section to determine what happens to (2/3)n and (3/2)n as n - > 
, and complete the question.
Have them attempt one or two more exercises, and then determine the formula, using 
, first with 
Students will need a brief review of the formulas for arithmetic series, as they were studied in Mathematics 10. It may be necessary to begin with one or two exercises in finding the sum of an arithmetic series before moving to word problems. Once the review is complete, students could work individually or in pairs on a set of word problems involving arithmetic and geometric series. The students should decide which type of series is inherent in the problem and what information is pertinent to the problem. Then the appropriate formula can be selected and the calculations carried out. Answers should be checked for reasonableness and solutions should be written.
These next two sections could be done in conjunction with the previous section. These problems should be addressed throughout the unit and are shown as real-world applications of geometric sequences. The students could work individually, in pairs or in small groups to solve a few of these types of problems. Once the initial set has been solved, students could work individually on a short assignment of other problems of this type.
Various problems could also be posed or elicited from the students.
Decide on an amount of money you would like to earn as a yearly salary in the next few years. What amount of money would you need at a rate of 7% to generate this yearly salary? Call this amount m. Suppose you wish to retire at age 55. In order to have a retirement income equal to your salary, you need to save amount m. How much would you have to save each year, assuming a rate of interest of 7%, compounded yearly, to equal amount m?
Students should use calculators or computers to help with the calculations. They can also utilize logs in the calculations.