This primer gives you a refresher on some basic algebra
concepts , such as solving
simultaneous equations , use of exponents and logs , and exp onential and
logarithmic
functions.
SOLUTION OF SIMULTANEOUS EQUATIONS
If we know two ways that two variables are related , we can solve for the
equilibrium value of
those variables. For instance, suppose that price p and quantity Q are by a
demand equation
and a supply equation
We can solve these equations simultaneously to find the values for p and Q where
demand equals
supply at equilibrium. One way of solving simultaneous equations is to use the
first equation to
solve for one variable and substitute it into the second equations. For example,
we can substitute
the expression for Q in the first equation, Q = a – bP, into the second to get:
a – bP = c + dP.
Now we can solve for P by collecting terms:
a – c = dP + bP, or
a – c = P(d + b).
Dividing through by (d + b), we get our equilibrium value of P:
Now we can substitute this ex pression for P * back into either the demand or
supply equation to
find the equilibrium expression for Q*. For example, subbing P* into the supply
equation gives
us:
So we now have equilibrium expressions for P* and Q* in
terms of only the parameters a, b, c,
and d. This is the point where the demand curve intersects the supply curve.
PROBLEMS:
1. Find the equilibrium solution for each of the following economic models of
supply and
demand
EXPONENTS AND LOGS, AND EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
Exponents: The term exponent indicates the power to which a number or
variable is raised. For
example, in the expression x3, which is read “x to the third power,”
the exponent is three.
Numerical examples:
Graphical examples :
Logarithms: Exponents are closely related to
logarithms. Consider two numbers such as 3 and
9 that can be related by the equation 32 = 9. Here we can define the
exponent 2 to be the
logarithm of 9 to the base of 3. We can write this new relationship as follows:
The base of the logarithm, which is 3 in the example
above, does not have to be restricted to any
particular number. But in many applications, two numbers are widely chosen as
bases – the
number 10 and the transcendental number e. Logarithms that have a base equal to
10 are called
common logarithms , or common “logs” for short.
Examples of common logs include:
Logarithms that have a base equal to e are called natural
logarithms, or natural “logs” for short.
Natural logs even have a special notation – ln instead of
:
Fore example, 
Before proceeding with natural logs, we need to learn more about the number e.
The transcendental number e has many special properties , just like another
famous
transcendental number, π (pronounced pi). One way the number π is
defined is by the
relationship of the circumference (C) or area (A) or a circle to its radius (r):
Circumference of a Circle: C = 2πr 
Area
of a circle: A = πr2
One way the number e is defined is through the following function:
For larger and larger values of x, the function f(x) will
get larger and larger, but by smaller
amounts each time. For example, if we let x start at 1 and get larger, we find
that
The transcendental number e is defined as x approaches
infinity (∞),
The numeric value of e is approximately 2.71828.
Examples of natural logarithms in include:
Rules of Logarithms
Rule I ( log of a product ): ln(xy) = ln x + ln y
Example 1: 
Example 2: 
Coutner Example!: ln(u + v) ≠ ln u + ln v
Rule II (log of a quotient): ln(x/y) = ln x - ln y
Example 3: 
Example 4: 
Rule III (log of a power): 
Example 3: 
Example 4: 
Example 5: 
Graphical Examples of log and exponential functions
Graphs of y = ex, and y = ln x :
PROBLEMS:
1. Evaluate the following:
2. Evaluate the following by application of the rules of
logarithms:
3. Which of the following are valid?
Economic Interpretation of the Transcendental Number e
Suppose we were lucky enough to find a bank willing to offer us the unusual
interest rate of 100
percent, compounded continuously. Then the number e = 2.71828 can be interpreted
as the yearend
value to which $1 will grow if interest at the rate of 100 percent per year were
compounded
continuously.
Compound interest formula:
What if we have an interest rate other than 100 percent? Suppose we have r =
0.05, or a 5
percent interest rate, and we start with principal of $A, and invest it for t
years. Finally, let’s
have m compounding period, so that if m=1, it’s only compounded annually and if
m=12, it’s
compounded monthly. The formula for the value , V(m) of our investment is depends
on m and
the interest rate r:
As the frequency of compounding increases, so does V(m).
To say that interest is compounded
continuously is that same as saying that m approaches infinity in the limit.
When this happens,
our formula becomes:
PROBLEM:
1. What’s the value of an $100 investment five years from now if the interest
rate is r = 0.06
and
(a) Interest is compounded annually?
(b) Interest is compounded monthly?
(c) Interest is compounded continuously?
SOLVING QUADRATICS-THE QUADRATIC FORMULA
For a quadratic, such as
ax2 + bx+c=0
where a,b, and c are constants, the values for x may be found by using the
quadratic formula:
Example:
