Math 142 Business Mathematics II
Minh Kha
Sections 1.3-1.5: Exponential and Logarithmic Functions
Goals: to study some exponential and logarithmic models in some applied fields.
1
Exponential functions
Definition 1.1. A generalized exponential function is a function of the form abx , where a is a real
number constant with a 6= 0 and the base, b, is a real number with b > 0 and b 6= 1.
Properties of General Exponential Functions:
• Domain: (−∞, ∞), i.e. all real numbers.
• Range: (0, ∞), i.e. all positive numbers.
• Graphs are continuous curves (no holes, gaps, or jumps).
• y = 0 is a horizontal asymptote in one direction.
• y-intercept: (0, a), i.e. all graphs pass through the point (0, a).
• If a > 0 and b > 1, then the graph increases as x increases (exponential growth function).
• If a > 0 and 0 < b < 1, then the graph decreases as x increases (exponential decay function).
Example 1. Classify the following functions as exponential growth or decay functions.
a) f (x) = (0.5)x
b) g(x) = 11(40)x
2 x
9
c) h(x) =
3
Example 2. Find the domain of f (x) = 3π x
−1/2
.
Proposition 1.2. Suppose that a and b are two positive real numbers such that a, b 6= 1. Let x, y be
any two real numbers. Then we have
• ax · ay = ax+y
• ax /ay = ax−y
• a−x = 1/ax
• (ab)x = ax · bx
• (a/b)x = ax /bx
• (ax )y = axy
1
• ax = ay if and only if x = y.
• a0 = 1
• For x 6= 0, ax = bx if and only if a = b.
2
A Mathematical Model of Compound Interest
Any positive number can be used as the base for an exponential function, but some bases are used more
frequently then others, e.g. 2, 10 and e.
Definition 2.1. The number e:
It can be shown that as n gets larger and larger, (1 +
1 n
) approaches the value 2.718281828... = e.
n
e is called the natural number.
The exponential function with base e has the form f (x) = aebx where a and b are real number constants.
Properties of Exponential Function with Base e: If a > 0 and b 6= 0,
• Domain: (−∞, ∞)
• Range: (0, ∞)
• y-intercept: (0, a)
• If b > 0, then it is an exponential growth function.
• If b < 0, then it is an exponential decay function.
4x
Example 3. Simplify: f (x) = (x2 +1)
2
Example 4. Solve the following equations:
a) 3x = 9x+1
b) x2 ex − x3 ex = 0
2
Growth and Decay
Definition 2.2. To model population growth and radioactive decay, it is often to use an exponential
function of the form
y = Cekt ,
where C is the initial amount, k is the relative growth (or decay) rate, and t is time.
If k is positive, it is exponential growth model, while if k is negative, it is exponential decay
model.
Example 5. The population of a town grows continuously at a relative growth rate of 5%. If 5,000
people currently live in the town, what will be the population in ten years?
Example 6. The population of an undesirable city is modeled by
p(t) = Ce−0.1t
where t represents the number of years since 1990, p(t) represents the population in the tth year, and C
is a constant representing the population in 1990.
a) If the city had a population of 50, 000 in 2000, what is the value of C?
b) Use the model to predict the population in 2100.
3
Compound Interest
Definition 2.3. The most common type of interest is compound interest, i.e., interest that is
calculated not only on the original investment amount but also on the previous interest earned, provided
that the time period is more than one period.
Suppose a principal P (present value) earns interest at the annual rate of r (expressed as a decimal),
and interest is compounded m times a year. Then the amount F (future value) after t years is
F = P (1 +
r mt
)
m
Here n = mt is the number of time periods.
Example 7. If 5, 000 dollars is invested in an account paying 5% compounded monthly, how much will
be in the account at the end of 5 years?
Continuously Compounded Interest
Definition 2.4. If a principal P earns interest at the annual rate of r (expressed as a decimal), and
interest is compounded continuously, then the amout F after t years is
F = P ert
Example 8. What amount will an account have after five years if 2, 000 is invested at an annual rate
of 5% compounded continuously?
Effective Yield
Motivation: If 1000$ is invested at an annual rate of 9% compounded monthly, then at the end of a
year there is
0.09 12
) = 1093.81$
F = 1000$ × (1 +
12
in the account. This is the same amount that is obtainable if the same principal of 1000 is invested for
one year at an annual rate of 9.381%. We call the rate 9.381% the effective annual yield ref f . The
9% annual rate is often refered to as the nominal rate.
In other words,
F = P (1 +
r m
) = P (1 + ref f ).
m
4
Thus,
(1 + ref f ) = (1 +
r m
) .
m
Definition 2.5. Assume a sum of money is invested at an annual rate of r expressed as a decimal and
is compounded m times a year. The effective yield ref f is
ref f = (1 +
r m
) − 1.
m
Similarly,
ref f = er − 1
is the effective annual yield compounded continuously.
Example 9. A bank advertised a nominal rate of 7.1% compounded semiannually. Another bank
advertises a nominal rate of 7% compounded continuously. What are the effective yields? In which
bank would you deposit your money?
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WE WILL USE TVM SOLVER. (We cannot use TVM Solver for simple interest calculations, only
compound interest.)
Compound Interest with TVM Solver on the Calculator:
1. Press APPS.
2. Select Finance... (Option 1) (Please note that if you have a plain TI-83, you need to press 2nd x
−1to access the Finance Menu).
3. Select TVM Solver (Option 1)
• N = total number of compoundings during the time period
• I% = interest rate (as a percent)
• P V = initial amount in the account
• P M T = regularly occuring payments
• F V = future value of the account
• P/Y = C/Y = the number of payments/compoundings per year
• END/BEGIN (Note: END should be higlighted)
4. Enter the known information.
5. Scroll to the line representing the unknown data.
6. Press SOLVE (ALPHA → ENTER)
Important Note: In the TVM Solver, the values for PV, PMT, and FV will sometimes be negative.
This is done to represent the transfer or flow of money. We will usually look at these problems from
the standpoint of the investor or borrower. e.g. a negative number represents an of money away from
the investor or borrower, i.e., when money is leaving your pocket. Use a negative number when:
• Making payments
• Depositing money in a bank
Use a positive number when:
• You receive a loan from a bank or lender.
• You receive money from a bank account.
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Example 10. How much is in an account after 30 years if $10,000 was initially deposited into an
account earning 10% per year compounded
a) annually
b) monthly
c) continuously
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3
Logarithms
Motivation: Every exponential function f (x) = ax , with a > 0, a 6= 1, is a one-to-one function and
thus, it has an inverse function. Then what is the inverse function of exponential function?
It is called the Logarithmic function with base a and is denoted loga .
Definition 3.1. Let a be a positive number with a 6= 1. Then the Logarithmic function with base
a, denoted by loga x, is defined as follows:
y = loga x ⇔ ay = x (x > 0).
Definition 3.2. There are some bases a that are important and convenient. We list here:
• Common Logarithm
The logarithm with base 10 is called the Common Logarithm and is denoted by omitting the
base
y = log x = log10 x ⇔ 10y = x
• Natural Logarithm
The logarithm with base e is called the Natural Logarithm and is denoted by ln
ln x = loge x (x > 0).
Note that the natural logarithm function y = ln x is the inverse of the exponential function y = ex ,
i.e.,
y = ln x ⇔ ey = x.
Example 11. Solve for x:
a) ln(e2x ) = x2
b) log36 ( 61 ) = x
Properites of Logarithms:
1. aloga x =
2. loga ax =
3. loga xy =
4. loga
x
=
y
8
5. loga xb =
6. loga b = loga c if and only if b = c.
1
Example 12. Simplify e 2 ln 81 .
Example 13. Write the given quantity in terms of log x, log y, and log z.
xy 2
log √ 7
( z)
Example 14. Write the given quantity as one logarithm.
11 log4 x − 23 log4 y
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Example 15. Solve for x.
√
a) 8
x
= 64
b) log (3x) = log (27x2 )
Example 16. How long will it take for the amount in an account to double if the money is compounded
continuously at an interest rate of 10% per year?
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Example 17. A bacteria culture starts with 10000 bacteria, and after 2 hours the population is 30000
bacteria. Assuming that the culture grows exponentially, find the population after 4 hours.
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