1. Exponential and Logarithmic Functions

2. 2-1 Exponential Functions
Function type
Example
linear
f(x) = 3x + 2
quadratic
f(x) = 3x2 + 2x - 3
polynomial
f(x) = 2x4 + 3x2
exponential
f(x) = 2x f(x) = 2.63x-1

3. Graph Exponential Functions (b > 1)
Graph y = 2x for x = -3 to 3 x y -3 -2 -1 1 2 3
1/8
1/4
1/2 1 2 4 8

4. Graph: f(x) = 2x

5. Graph: f(x) = 2-x

6. Graph Exponential Function (0< b < 1)
Graph y = (1/2)x for x = -3 to 3 x y -3 -2 -1 1 2 3 8 4 2 1
1/2
1/4
1/8

7. 2 is the base of the exponential –“exponential base 2”
the base can be any positive number
common bases are 2, 10, and e
Exponential functions with base e
f(x) = ex
e is a real number constant (like ) value = …
frequently seen as the base for exponential functions
called the natural base

8. Properties of the exponential functions:
f(x) = bx and f(x) = b-x, - < x < 
b is called the base
can always make b > 1
e.g. f(x) =
= 2-x
domain: (-, ) range: (0, )
continuous
f(x) = bx is an increasing function
f(x) = b-x is a decreasing function

9. y-intercept: y = 1
no x-intercepts; graph always above x axis
x-axis is an asymptote:
as x  +  for f(x) = b-x
as x  -  for f(x) = bx
bx = by  x = y (one-to-one property)

10. Growth
Decay

11. Exponential function – A function of the form y=abx, where b>0 and b1.
Step 1 – Make a table of values for the function.

12. Make the Graph
Now that you have a data table of ordered pairs for the function, you can plot the points on a graph.
(-2, 1/9) (0,1) (2,9)
Draw in the curve that fits the plotted points. y x y x

13. APPLICATION
Simple Interest formulas
I = Prt
A = P + Prt = P(1 + rt)

14. P = principal invested
(also called present value)
r = annual interest rate
(expressed as a decimal)
t = time in years
I = interest earned
A = total amount after t years
(also called future value)

15. Simple Interest
You invest $ for at 10% simple interest.
How much do you have at the end of 2 years?
We can do this in our heads:
10% of $100 is $10
that's $10 interest earned per year
for 2 years for a total of $20 interest
plus the $100 original investment . . .
for a new amount of : $120

16. P = ($100)
r = (10%)
t = (2 years)
I = ($20)
A = ($120.00)

17. If you know any 3 of the variables, the formula (plus some algebra)
can be used to solve for the 4th variable:
Example:
$100 is invested (simple interest) for 10 years, and the investment doubled in value.
What was the interest rate?
The equation:
200 = 100(1 + r(10))
Solve: r = 0.10= (10%)

18. Example: Compound Interest
You deposit $1500 in an account that pays 2.3% interest compounded yearly,
What was the initial principal (P) invested?
What is the growth rate (r)? The growth factor?
Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money?
The initial principal (P) is $1500.
The growth rate (r) is The growth factor is

19. 2-2 Logarithmic Functions
The common log of a number is that exponent (or power) to which 10 must be raised to obtain the number.
Notation: y = log (x) or y = log x
"y = log of x"
Example:
log (1000) =
. . . the power to which 10 must be raised to obtain 1000
so log(1000) = 3

20. Another way to put it: 3 = log(1000) because 103 = 1000
y = log(x) means 10y = x
Be CAREFUL! log x + 2  log(x + 2)
log x + 2 = log(x) + 2

21. Graphing a log function
function y = log2 x: x 1 4 8
1/2 y 2 3 -1

23. Recall: if m = 102 then 2 = log m 100000 = 100 x 1000 mn = m x n
Properties of logs
Property 1
Recall: if m = 102 then 2 = log m
= x 1000
mn = m x n
= x 103
10log mn = 10log m x 10log n
10log mn = 10log m + log n (laws of exponents)
log mn = log m + log n (if 10x = 10y then x = y)
the log of a product = the sum of the logs 23

24. =============================== 100 = 100000  1000 m/n = m  n
Property 2
===============================
=  1000
m/n = m  n
=  103
10log m/n = 10log m  10log n
10log m/n = 10log m - log n (laws of exponents)
log m/n = log m - log n (if 10x = 10y then x = y)
the log of a quotient = the difference of the logs
========================================== 24

25. log mr = log mm … m (for r factors)
Property 3
log mr = log mm … m (for r factors)
= log m + log m log m (for r terms)
= r log m
log mr = r log m
the log of a power = the exponent times the log of the base 25

26. These relationships hold for any base:
1. loga mn = loga m + loga n (log of a product)
2. loga m/n = loga m - loga n (log of a quotient)
3. loga mr = r loga m (log of a power)
Each property can be used in two directions, e.g.
log (10)(20) = log 10 + log 20
uses property 1 going from left-hand to right-hand side
called expansion () 26

27. 3 log x not acceptable as an answer, but log x3
log log 20 = log (10/20)
uses property 2 going from right-hand to left-hand side
called collection, or writing as the log of a single expression “log expression” ()
the book directions are “write as a one logarithm” - this is ambiguous. Read “Write as the log of a single expression”
3 log x not acceptable as an answer, but log x3 27

28. Examples: Write Equivalent Equations
Examples: Write the equivalent exponential equation
and solve for y.
Solution
Equivalent Exponential Equation
Logarithmic Equation
y = log216
16 = 2y
16 = 24  y = 4
= 2 y
= 2-1 y = –1
y = log2( )
y = log416
16 = 4y
16 = 42  y = 2
y = log51
1 = 5 y
1 = 50  y = 0
Examples: Write Equivalent Equations

29. Properties of Logarithms
1. loga 1 = 0 since a0 = 1.
2. loga a = 1 since a1 = a.
3. loga ax = x and alogax = x inverse property
4. If loga x = loga y, then x = y. one-to-one property
Examples: Solve for x: log6 6 = x
log6 6 = 1 property 2 x = 1
Simplify: log3 35
log3 35 = 5 property 3
Simplify: 7log79
7log79 = 9 property 3
Properties of Logarithms

30. Properties of Natural Logarithms
1. ln 1 = 0 since e0 = 1.
2. ln e = 1 since e1 = e.
3. ln ex = x and eln x = x inverse property
4. If ln x = ln y, then x = y. one-to-one property
Examples: Simplify each expression.
inverse property
inverse property
property 2
property 1
Properties of Natural Logarithms