Unit 3
Exponential and Logarithmic Functions
In the previous unit, we discussed algebraic functions, their domains, ranges and
graphs. For the next two units, we will be looking at functions which are not algebraic,
called transcendental functions. For this unit, we will consider a pair of one-to-one
transcendental functions, which are inverses of each other. These are the exponential
and logarithmic functions. We will also consider their domains, ranges and graphs. In
Section 3, we will go back to solving equations. This time, the equations will involve
exponential and logarithmic expressions. In the last section, we will also have a glimpse
of another set of transcendental functions, called the hyperbolic functions.
In this Unit, we aim to do the following:
1.
Identify properties and sketch graphs of exponential functions,
2.
Identify properties and sketch graphs of logarithmic functions,
3.
Solve equations involving exponential and logarithmic functions,
4.
Perform operations on hyperbolic and inverse hyperbolic functions, and
5.
Solve verbal problems involving exponential and logarithmic functions.
3.1 Exponential Functions: Properties and Graphs
Consider this problem: An amount of PhP 5,000 is deposited at 6% interest
compounded annually. Find the total amount after 3 years.
Let us observe at what happens to the amount invested in the table below.
Year
Amount
invested
Interest
Earned
Total
Amount
1
5000
300
5300
2
5300
318
5618
3
5618
337.08
5955.08
Note that every year the amount invested increases because of the interest earned in
that year. Thus, every year the interest earned also increases.
Now, let us give an
expression that will give us the total amount in
t
years, where
t
is any natural number.
If
P
is the amount invested and
i
is the interest rate (in decimal form), then we have
.
In this expression, the variable for time is the exponent, unlike in the previous
expressions that we have seen where we have constants as exponents. An expression
like this is called an exponential expression.