1. 3.1 Exponential Functions
– any function whose equation contains a variable in the exponent.
[measures rapid increase or decrease (Example: epidemic growth)]
f(x) = bx
f – exponential function b - constant base (b > 0, b  1) x = any real number -- domain is (-∞ , ∞)
f(x) = 2x g(x) = 10x h(x) = 3x+1
Shift up c units
f(x) = bx + c
Shift down c units
f(x) = bx – c
Shift left c units
f(x) = bx+c
Shift right c units
f(x) = bx-c
f (x) = 3x
g(x) = 3x+1
(0, 1)
(-1, 1) 1 2 3 4 5 6 -5 -4 -3 -2 -1
Graphing Exponential Functions:
32+1 = 33 = 27
32 = 9 2
31+1 = 32 = 9
31 = 3 1
30+1 = 31 = 3
30 = 1
3-1+1 = 30 = 1
3-1 = 1/3 -1
3-2+1 = 3-1 = 1/3
3-2 = 1/9 -2
g(x) = 3x+1
f (x) = 3x x

2. See P.193 All properties above are the same
Except #4 which is ‘decreasing’

3. 3.2 Investment and Interest Examples
Compound Interest
Karen Estes just received an inheritance of $10000
And plans to place all money in a savings account
That pays 5% compounded quarterly to help her son
Go to college in 3 years. How much money will be
In the account in 3 years?
Use the formula:
A = P(1 + r/n)nt
Simple Interest
Juanita deposited $8000
in a bank for 5 years at a
simple interest rate of 6%
1. How much interest will she get?
2. How much money will she have
at the end of 5 years
Use the formula:
I = Prt
I = (8000)(.06)(5)
= $2400 (interest earned in 5 years)
In 5 years she will have a total of
A = P + I
= = $10,400
A = amount in account after t years
P = principal or amount invested
t = time in years
r = annual rate of interest
n = number of times compounded
per year
Annual => n = 1
Semiannual => n = 2
Quarterly => n = 4
Monthly => n = 12
Daily => n = 365
As n increases (n  ∞), A does NOT increase indefinitely.
It gets closer and closer to a fixed number “e”, and gives us
the continuous compound interest formula: A = Pert

4. Compound Interest related to Simple Interest
If a principal of P dollars is borrowed for a period of
t years at a perannum interest rate r, the interest I charged is:
I = Prt
Annually -> Once per year Semiannually -> Twice per year
Quarterly -> Four times per year Monthly -> 12 times per year
Daily -> 365 times per year
Example: A credit union pays interest of 8% per annum compounded quarterly on a
Certain savings plan. If $1000 is deposited in such a plan and the interest is
Left to accumulate, how much is in the account after 1 year?
I = (1000)(.08)(1/4) = $20 => New principal now is: $1020
I = (1020)(.08)(1/4) = $20.40 => New principal now is: $
I = ( )(.08)(1/4) = $20.81 => New principal now is: $
I = ( )(.08)(1/4) = $21.22 => New principal now is: $
Compound Interest Formula: The amount, A, after t years due to a principal P
Invested at an annual interest rate r compounded n times per year is:
A = P  (1 + r/n)nt

5. The Natural Base e
An irrational number, symbolized by the letter e, appears as the base in many applied exponential functions. This irrational number is approximately equal to More accurately,
The number e is called the natural base. The function f (x) = ex is called the natural exponential function.
Analogous to the continuous
compound interest formula, A = Pert
The model for continuous growth and decay is: A(t) = A0ekt
A(t) = the amount at time t
A0 = A(0), the initial amount
k = rate of growth (k>0) or decay (k < 0)
In year 2000, population of the world was about 6 billion; annual rate of growth 2.1%. Estimate the population in 2030 and 1990.
2000=year 0, 2030=year 30, year 1990=year -10
A(30) = 6e(.021)(30) = (billion people)
A(-10) = 6e(.021)(-10) = (billion people)
** Note these are just estimates. The actual
population in 1990 was 5.28 billion
f (x) = ex
f (x) = 2x
f (x) = 3x
(0, 1)
(1, 2) 1 2 3 4
(1, e)
(1, 3) -1

6. Exponential Growth & Decay (Law of Uninhibited Growth/Decay)
A0 is the original amount of ‘substance’ (drug, cells, bacteria, people, etc…)

7. Solving Exponential Equations
e2x-1 = e-4x
e3x
e2x-1 = e-7x
2x – 1 = -7x
-1 = -9x
X = 1/9

8. 3.3 Logarithmic Functions
A logarithm is an exponent such that for b > 0, b  1 and x > 0
y = logb x if and only if by = x
Logarithmic equations Corresponding exponential forms
2 = log5 x 1) 52 = x
3 = logb ) b3 = 64
log3 7 = y 3) 3y = 7
y = loge 9 4) ey = 9
log25 5 = 1/2 because 251/2 = 5.
25 to what power is 5?
log25 5
log3 9 = 2 because 32 = 9.
3 to what power is 9?
log3 9
log2 16 = 4 because 24 = 16.
2 to what power is 16?
log2 16
Logarithmic Expression Evaluated
Question Needed for Evaluation
Evaluate the
Logarithmic Expression

9. Logarithmic Properties
Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).
Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
logb bx = x The logarithm with base b of b raised to a power equals that power.
b logb x = x b raised to the logarithm with base b of a number equals that number.
Graphs of f (x) = 2x and g(x) = log2 x [Logarithm is the inverse of the exponential function] 4 2 8 1
1/2
1/4
f (x) = 2x 3 -1 -2 x 2 4 3 1 -1 -2
g(x) = log2 x 8
1/2
1/4 x
Reverse coordinates.
Properties of f(x) = logb x
Domain = (0, ∞)
Range = (-∞, +∞)
X intercept = 1 ; No y-intercept
Vertical asymptote on y-axis
Decreasing on 0<b<1; increasing if b>1
Contains points: (1, 0), (b, 1), (1/b, -1)
Graph is smooth and continuous -2 -1 6 2 3 4 5
f (x) = 2x
f (x) = log2 x
y = x

10. Common Logs and Natural Logs
A logarithm with a base of 10 is a ‘common log’
log = ______ because 103 = 1000
If a log is written with no base it is assumed to be 10.
log 1000 = log = 3 3
A logarithm with a base of e is a ‘natural log’
loge 1 = ______ because e0 = 1
If a log is written as ‘ln’ instead of ‘log’ it is a natural log
ln 1 = loge 1 = 0

11. Doubling Your Investments
How long will it take to double your
Money if it earns 6.5% compounded
continuously?
Recall the continuous compound interest formula: A = Pert
If P is the principal and we want P to double, the amount A will be 2P.
2P = Pe(.065)t
2 = e(.065)t
Ln 2 = ln e(.065)t
Ln 2 = .065t
T = ln 2 = ≈ ≈ 11 years
At what rate of return compounded continuously would your money double in 5 years?
A = Pert
2P = Per(5)
2 = er(5)
ln 2 = ln e5r
.6931 = 5r
r = .6931/5
R = .1386
So, annual interest rate
needed is: 13.86%

12. Newton’s Law of Cooling
T = Ts + (T0 – Ts)e-kt T = Temperature of object a time t
Ts = surrounding temperature
T0 = T at t = 0
A McDonald’s franchise discovered that when coffee is poured from a CoffeeMaker
whose contents are 180° F into a noninsulated pot, after 1 minute, the Coffee cools to
165° F if the room temperature is 72° F. How long should employees wait before
pouring coffee from this noninsulated pot into cups to deliver it to customers at 125°F?
T0 = 180 and Ts = Now, T = e-.1495t
T = 72 + (180 – 72) e-kt = e-.1495t
T = e-kt = 108e-.1495t
.4907 = e-.1495t
Since T = 165 when t = ln = ln e-.1495t
165 = e-k(1) = t
93 = 108e-k t = 4.76
93 = 108e-k Employees should wait about 5 min
Ln = ln e-k to deliver the coffee at the desired
= -k temperature.
K = .1495

13. 3.4 Properties & Rules of Logarithms
Basic Properties
Logb b = 1 1 is the exponent to which b must be raised to obtain b. (b1 = b).
Logb 1 = 0 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
Inverse Properties
logb bx = x The logarithm with base b of b raised to a power equals that power.
b logb x = x b raised to the logarithm with base b of a number equals that number.
For M>0 and N > 0
Product Rule
logb(MN) = logb M + logb N
Quotient Rule
logb M = logb M - logb N N
Power Rule p
logb M = p logb M

14. Logarithmic Property Practice
Quotient Rule
Product Rule
logb M = logb M - logb N N
logb(MN) = logb M + logb N
log3 (27 • 81) =
2) log (100x) =
3) Ln (7x) =
1) log x
2) Ln e5 11 = =
Power Rule
log5 74
Log (4x)5
Ln x =
Ln x =
logb M = p logb M p = =

15. Expanding Logarithms logb(MN) = logb M + logb N
logb M = logb M - logb N N
logb M = p logb M p
1) Logb (x2 y )
3) log5 x
25y3 4)
log2 5x2 3
2) log6 3 x
36y4

16. Condensing Logarithms
logb(MN) = logb M + logb N
logb M = logb M - logb N N
logb M = p logb M p
Note: Logarithm coefficients
Must be 1 to condense.
(Use power rule 1st)
log4 2 + log ) 2 ln x + ln (x + 1)
Log 25 + log 4 5) 2 log (x – 3) – log x
Log (7x + 6) – log x 6) ¼ logb x – 2 logb 5 – 10 logb y

17. The Change-of-Base Property
Example: Evaluate log3 7
Most calculators only use:
Common Log [LOG] (base 10)
Natural Log [LN] (base e)
It is necessary to use the change
Of base property to convert to
A base the calculator can use.

18. Matching Data to an Exponential Curve
Find the exponential function of the form f(x) = aebx that passes through the
points (0,2) and (3,8)
F(0) = 2 f(3) = 8
2 = aeb(0) = aeb(3) f(x) = 2e((ln4)/3)x
2 = ae = 2e3b
2 = a(1) = e3b
a = 2 ln 4 = ln e3b
ln 4 = 3b
b = (ln4)/3

19. 3.5 Exponential & Logarithmic Equations
Step 1: Isolate the exponential expression.
Step 2: Take the natural logarithm on both sides of the equation.
Step 3: Simplify using one of the following properties:
ln bx = x ln b or ln ex = x.
Step 4: Solve for the variable using proper algebraic rules.
Examples (Solve for x):
log4(x + 3) = 2.
log 2 (3x-1) = 18
54x – 7 – 3 = 10
3x+2-7 = 27

20. More Equations to Try