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5.1 Exponential Functions

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a. y = yobt/h. y = 50 (2)(14/3) y = 1269 bacteria. 5.1 Exponential Functions ... b. y = yobt/h. 3000 = 50 (2)(t/3) 60 = 2t/3. 5.2 Simple and Compound Interest ... – PowerPoint PPT presentation

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Title: 5.1 Exponential Functions


1
5.1 Exponential Functions
Rules For Exponents If a 0 and b 0, the
following hold true for all real numbers x and y.
2
5.1 Exponential Functions
If we apply the quotient rule, we get
3
5.1 Exponential Functions
  • For any nonzero number x

and
4
5.1 Exponential Functions
  • Examples

5
5.1 Exponential Functions
Examples
  • (5x-2)3 125x-6125/x6
  • (3x/y3)2 9x2/y6
  • (4x)-1 1/(4x)
  • (2a3b-3c4)3 8a9b-9c12
  • 40 1
  • 2-1 ½
  • (½)-2 4
  • 5-2 1/25

6
5.1 Exponential Functions
Simplify
7
5.1 Exponential Functions
8
5.1 Exponential Functions
Simplify
Rewrite
Notice
9
5.1 Exponential Functions
Simplify
Rewrite
Notice
10
5.1 Exponential Functions
Simplify
Rewrite
Notice
11
5.1 Exponential Functions
  • Then
  • If

Examples
  • Since
  • Since
  • Since

12
5.1 Exponential Functions
In general, if n is a multiple of m, then
13
5.1 Exponential Functions
Use the rules for exponents to solve for x
  • 4x 128
  • (2)2x 27
  • 2x 7
  • x 7/2
  • 2x 1/32
  • 2x 2-5
  • x -5

14
5.1 Exponential Functions
  • 27x 9-x1
  • (33)x (32)-x1
  • 33x 3-2x2
  • 3x -2x 2
  • 5x 2
  • x 2/5
  • (x3y2/3)1/2
  • x3/2y1/3

15
5.1 Exponential Functions
Definition Exponential Function Let a be a
positive real number other than 1, the function
f(x) ax is the exponential function with base
a.
16
5.1 Exponential Functions
y 2 x
  • If b 1, then the graph of b x will
  • Rise from left to right.
  • Not intersect the x-axis.
  • Approach the x-axis.
  • Have a y-intercept of (0, 1)

17
5.1 Exponential Functions
y (1/2) x
  • If 0
  • Fall from left to right.
  • Not intersect the x-axis.
  • Approach the x-axis.
  • Have a y-intercept of (0, 1)

18
5.1 Exponential Functions
Natural Exponential Function where e is the
natural base and e ? 2.718
19
5.1 Exponential Functions
(-8, 8)
(-8, 8)
(-8, 8)
(0, 8)
(0, 8)
(0, 8)
Dec.
Inc.
Inc.
(0, 1)
20
5.1 Exponential Functions
Use translation of functions
to graph the following.
Determine the domain and range f (x) 2(x
2) 3
Domain (-8, 8) Range (-3, 8)
21
5.1 Exponential Functions
Definitions Exponential Growth and Decay
The function y k ax, k 0 is a model for
exponential growth if a 1, and a model for
exponential decay if 0 y new amount yO original amount b
base t time h half life
22
5.1 Exponential Functions
  • An isotope of sodium, Na, has a half-life of 15
    hours. A sample of this isotope has mass 2 g.
  • Find the amount remaining after t hours.
  • Find the amount remaining after 60 hours.
  • b. y yobt/h
  • y 2 (1/2)(60/15)
  • y 2(1/2)4
  • y .125 g
  • a. y yobt/h
  • y 2 (1/2)(t/15)

23
5.1 Exponential Functions
  • A bacteria double every three days. There are 50
    bacteria initially present
  • Find the amount after 2 weeks.
  • When will there be 3000 bacteria?
  • a. y yobt/h
  • y 50 (2)(14/3)
  • y 1269 bacteria

24
5.1 Exponential Functions
A bacteria double every three days. There are 50
bacteria initially present When will
there be 3000 bacteria?
  • b. y yobt/h
  • 3000 50 (2)(t/3)
  • 60 2t/3

25
5.2 Simple and Compound Interest
Formulas for Simple Interest Suppose P dollars
are invested at a simple interest rate r, where r
is a decimal, then P is called the principal and
P r is the interest received at the end of one
interest period.
26
5.2 Simple and Compound Interest
Formulas for Compound Interest After t years,
the balance A in an account with principal P and
annual interest rate r is given by the two
formulas below.
1. For n compoundings per year
2. For continuous compounding
27
5.2 Simple and Compound Interest
Find the balance after 10 years if 1000.00 is
invested at 4 and the account pays simple
interest.
28
5.2 Simple and Compound Interest
Find the balance after 10 years if 1000.00 is
invested at 4 and the interest is compounded
1485.95
a. Semiannually
b. Monthly
1490.83
c. Continuously
1491.82
29
5.3 Effective Rate and Annuities
Effective Annual Rate The effective annual rate
of ieff of APR compounded k times per year is
given by the equation Another name for
effective annual rate is effective yield
30
5.3 Effective Rate and Annuities
What is the better rate of return, 7 compounded
quarterly or 7.2 compounded semianually?
31
5.3 Effective Rate and Annuities
1.071 1 .071 7.1
1.073 1 .073 7.3
7.2 compounded semiannually is better.
32
5.3 Effective Rate and Annuities
What is the better rate of return, 8 compounded
monthly or 8.2 compounded quarterly?
33
5.3 Effective Rate and Annuities
8.3
8.5
8.2 quarterly is better.
34
5.3 Effective Rate and Annuities
Future Value of an Ordinary Annuity The Future
Value S of an ordinary annuity consisting of n
equal payments of R dollars, each with an
interest rate i per period is
35
5.3 Effective Rate and Annuities
Suppose 25.00 per month is invested at 8
compounded quarterly. How much will be in the
account after one year?
  • 1st quarter 25.00
  • 2nd quarter 25.00(1.08/4) 25.00 50.50
  • 3rd quarter 50.50(1.08/4) 25.00 76.51
  • 4th quarter 76.51(1.08/4) 25.00 103.04

36
5.3 Effective Rate and Annuities
Present Value of an Ordinary Annuity The Present
Value A of an ordinary annuity consisting of n
equal payments of R dollars, each with an
interest rate i per period is
37
5.4 Logarithmic Functions
The inverse of an exponential function is called
a logarithmic function.
Definition x a y if and only if y log a x
38
5.4 Logarithmic Functions
  • log 4 16 2 ? 42 16
  • log 3 81 4 ? 34 81
  • log10 100 2 ? 102 100

39
5.4 Logarithmic Functions
Sketch a graph of f (x) 2x and sketch a graph
of its inverse. What is the domain and range of
the inverse of f.
Domain (0, 8) Range (-8, 8)
40
5.4 Logarithmic Functions
The function f (x) log a x is called a
logarithmic function.
  • Domain (0, 8)
  • Range (-8, 8)
  • Asymptote x 0
  • Increasing for a 1
  • Decreasing for 0
  • Common Point (1, 0)

41
5.4 Logarithmic Functions
Find the inverse of g(x) 3x.
Note The function and its inverse are
symmetrical about the line y x.
42
5.4 Logarithmic Functions
Find the inverse of g(x) ex.
ln x is called the natural logarithmic function
43
5.4 Logarithmic Functions
So
So
So
So
44
5.4 Logarithmic Functions
  • loga(ax) x for all x ? ?
  • alog ax x for all x 0
  • loga(xy) logax logay
  • loga(x/y) logax logay
  • logaxn n logax

Common Logarithm log 10 x log x Natural
Logarithm log e x ln x All the above
properties hold.
45
5.4 Logarithmic Functions
Product Rule
46
5.4 Logarithmic Functions
Quotient Rule
47
5.4 Logarithmic Functions
Power Rule
48
5.4 Logarithmic Functions
Expand
49
5.4 Logarithmic Functions
Find an equation of best fit for the data (1,3),
(2,12), (3,27), (4,48)
50
5.5 Graphs of Logarithmic Functions
The function f (x) log a x is called a
logarithmic function.
  • Domain (0, 8)
  • Range (-8, 8)
  • Asymptote x 0
  • Increasing for a 1
  • Decreasing for 0
  • Common Point (1, 0)

51
5.5 Graphs of Logarithmic Functions
The natural and common logarithms can be found on
your calculator. Logarithms of other bases are
not. You need the change of base formula.
where b is any other appropriate base. (usually
base 10 or base e)
52
5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (2,?) Range (-?, ?)
53
5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-2,?) Range (-?, ?)
54
5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-3,?) Range (-?, ?)
55
5.5 Graphs of Logarithmic Functions
On the Richter scale, the magnitude R of an
earthquake can be measured by the intensity
model.
R Magnitude a Amplitude T Period B
Damping Factor
56
5.5 Graphs of Logarithmic Functions
What is the magnitude on the Richter scale of
an earthquake if a 300, T 30 and B 1.2?
57
5.6 Solving Exponential Equations
  • Solve 4 3x 16 x 2
  • The bases can be rewritten as
  • (22) 3x (24) (x 2)
  • 2 6x 2 4x 8
  • 6x 4x 8
  • 2x -8
  • x -4

58
5.6 Solving Exponential Equations
  • To solve exponential equations, pick a convenient
    base (often base 10 or base e) and take the log
    of both sides.
  • Solve

59
5.6 Solving Exponential Equations
  • Take the log of both sides
  • Power rule

60
5.6 Solving Exponential Equations
  • Solve for x
  • Divide

61
5.6 Solving Exponential Equations
  • To solve logarithmic equations, write both sides
    of the equation as a single log with the same
    base, then equate the arguments of the log
    expressions.
  • Solve

62
5.6 Solving Exponential Equations
  • Write the left side as a single logarithm

63
5.6 Solving Exponential Equations
  • Equate the arguments

64
5.6 Solving Exponential Equations
  • Solve for x

65
5.6 Solving Exponential Equations
66
5.6 Solving Exponential Equations
  • Check for extraneous solutions.

67
5.6 Solving Exponential Equations
  • To solve logarithmic equations with one side of
    the equation equal to a constant, change the
    equation to an exponential equation
  • Solve

68
5.6 Solving Exponential Equations
  • Write the left side as a single logarithm

69
5.6 Solving Exponential Equations
  • Write as an exponential equations

70
5.6 Solving Exponential Equations
  • Solve for x

71
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