ELEMENTARY & INTERMEDIATE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: This material is extensively elaborated upon in my optional e-book GOLDen Mathematics: Intermediate Algebra. This section of material only is downloadable for nominal fee at www.lulu.com/content/378815. See "Tell me more about Keely's GM book".
Exponential Functions and Graphs
GOLDen Mathematics - Intermediate Algebra: Section 11.1
Supplemental Sites: MathOL Links - Alg 11.1
Topics of Importance
Exponential function: definition, terminology,
evaluate
Graphs of exponential functions: basic shape, domain, range, intercepts,
asymptotes, reflections, and shifts
e: definition, evaluate on calc
Comments and Cautions
We have studied linear and quadratic functions both of
which are simple polynomial functions. But not all functions are polynomial.
Today we will be investigating exponential functions. These functions have tons
of real-world applications! For instance, you've heard of diseases like AIDS
spreading "exponentially"? Spread of diseases, population growth, compounding
interest, and carbon dating are all examples of applications that can be modeled
using exponential functions. This is pretty cool stuff! Keep your graphing
calculator handy as you explore exponential functions such as y=2x
(note the variable is in the power not the base). Watch what effect the
following things have on the graph: a negative in front, different bases, adding
or subtracting a number from the basic function, adding or
subtracting a number to the x in the power. Watch for the
effects on the shape, axis intercept points, asymptote line(s), domain, and
range.
I think it is useful to introduce the scientific constant "e" at this time too. It is defined in chapter 11.5 of your text. "e" is an irrational number like π except that e is equal to approximately 2.718. Try graphing y=ex and variations (reflections, shifts, translations) of this function. How does it compare with the graph of y=2x and y=3x? Why?
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 12.1.)
 | ch 12.1 pg 807 The content of the "enrichment essay" can be skipped as it will be covered in a college algebra course instead.ch 11.2 pg 855 The graph at the bottom of the page shows y=2^x and x=2^y. Note that they are mirror images of one another about the line y=x. In the next section we will learn that the graph of x=2^y is more commonly written as the logarithm y=log2x. |
Logarithmic Functions and Graphs
GOLDen Mathematics - Intermediate Algebra: Section 11.2
Supplemental Sites: MathOL Links - Alg 11.2
Topics of Importance
Logarithmic function: definition, common
logarithm, natural logarithm
Graphs of logarithmic functions: basic shape, domain, range, intercepts,
asymptotes
Evaluate logarithms (by hand and on calc), change of base theorem
Convert logarithmic 
exponential form
Properties of logarithms: basic properties & product, quotient, power rules
Comments and Cautions
After studying exponential functions, it seems a natural
question to ask, what is the inverse of an exponential function? What function
"undoes" say 2x enabling you to solve for x in the equation 2x=5
for instance? Well, a logarithmic function is the inverse of an exponential
function. These special functions are the focus of our work today. Logarithms
such as log3x (read "log base 3 of x") have their own graphs and
algebraic rules. Be sure to memorize the "properties of logarithms" and the
"rules of logarithms" which define the operations that can be conducted on
logarithms as we enter a whole new algebra! One caution: These rules of
logarithms show that, for instance, log(2x) does not equal log(2)*log(x)
and log(x+2) does not equal log(x)+log(2). Be careful how you work with
logarithms when simplifying logarithmic expressions. Be sure to follow the
rules! Lastly, be sure to study both the algebraic processes, the connection
back to exponential functions, and evaluating logarithms on the calculator
(including using the "change of base theorem"). It is also important to know the
difference between the "common logarithm" and the "natural logarithm".
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 12.2-12.3.)
| ch 12.2 pg 813 The conversion from logarithmic to exponential form (and visa versa) is particularly important to learn. Try to keep in mind that 23=8 and log28=3 say the same thing just in different forms (exponential form and logarithmic form, respectively). |
| ch 12.2 pg 819-820 Take note of the basic properties of common and natural logarithms given in the two charts. |
| ch 12.3 pg 828-829 Take note of the rules of logarithms (the product rule, the quotient rule, and the power rule), in both directions "expanding" and "condensing", given in the two charts. |
| ch 12.3 pg 828 Check out the side bar "study tip" box! It cautions you against some very common errors that can occur when simplifying logarithms. |
| ch 12.3 pg 831 It is recommended that you memorize the change of base theorem,
it will come in handy over and over. Note that it can be
written as logbx = log(x)/log(b) or as logbx = ln(x)/ln(b)
with the latter being more common. Realize that it can be used to enter
logarithms on your calculator. For instance, log415 can be entered
as ln(15)/ln(4)  1.853 (try it). |
| pg 863-864 has a nice overview of all the properties and rules of logarithms. |
Solving Exponential and Logarithmic Equations
GOLDen Mathematics - Intermediate Algebra: Section 11.3
Supplemental Sites: MathOL Links - Alg 11.3
Topics of Importance
Solve logarithmic equations algebraically: by converting to
exponential form, via properties of logs
Solve exponential equations algebraically: having the same base, having
different bases
Solve graphically
Applications of exponential and logarithmic functions
Comments and Cautions
Wow, our last section! Here we solve equations that contain exponential or
logarithmic expressions. Concentrate on the basic equations. If you take a
pre-calculus class then time will be spent there on more complicated and
involved equations. The applications of exponential and logarithmic functions
are extensive, but again, we can work on these more in a pre-calculus class.
Give yourself some brief exposure to the applications mostly as a motivating
factor, but don't let them side-track you from the solving of basic equations.
Focus on knowing how to solve each of the basic styles of equations and which
processes to use on which equation.
Text Notes (These notes refer to Introductory & Intermediate Algebra for College Students 2nd ed by Blitzer sections 12.4-12.5.)
| ch 12.4 is a really important section. Be sure to try a variety of problems so you get to solve using several different methods! |
| ch 12.5 Cover pages 848-853 ONLY stopping at the point that it says, "The Art of Modeling". SKIP the remaining pages including examples 3-5. This later material is best suited for a college algebra class. You will only be tested on the "population growth" and/or "carbon dating" type problems in this section. |
| And on that note we conclude the course content! Woo-hoo!! |
Originally written: 2006-006-15
Last revision:
2008-03-01 03:08 AM