5.1 Exponential Functions

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5.1 Exponential Functions
Rules For Exponents If a gt 0 and b gt 0, the
following hold true for all real numbers x and y.
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5.1 Exponential Functions
If we apply the quotient rule, we get
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5.1 Exponential Functions

• For any nonzero number x

and
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5.1 Exponential Functions

• Examples

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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

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5.1 Exponential Functions
Simplify
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5.1 Exponential Functions
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5.1 Exponential Functions
Simplify
Rewrite
Notice
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5.1 Exponential Functions
Simplify
Rewrite
Notice
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5.1 Exponential Functions
Simplify
Rewrite
Notice
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5.1 Exponential Functions

• Then

• If

Examples

• Since

• Since

• Since

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5.1 Exponential Functions
In general, if n is a multiple of m, then
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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

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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

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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.
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5.1 Exponential Functions
y 2 x

• If b gt 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)

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5.1 Exponential Functions
y (1/2) x

• If 0 lt b lt 1, then the graph of b x will
• Fall from left to right.
• Not intersect the x-axis.
• Approach the x-axis.
• Have a y-intercept of (0, 1)

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5.1 Exponential Functions
Natural Exponential Function where e is the
natural base and e ? 2.718
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5.1 Exponential Functions
Function f(x) 2x h(x) (0.5)x g(x) ex
Domain
Range
Increasing or Decreasing
Point Shared On All Graphs
(-8, 8)
(-8, 8)
(-8, 8)
(0, 8)
(0, 8)
(0, 8)
Dec.
Inc.
Inc.
(0, 1)
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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)
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5.1 Exponential Functions
Definitions Exponential Growth and Decay
The function y k ax, k gt 0 is a model for
exponential growth if a gt 1, and a model for
exponential decay if 0 lt a lt 1.
y new amount yO original amount b
base t time h half life
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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)

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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

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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

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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.
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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
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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.
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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
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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
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5.3 Effective Rate and Annuities
What is the better rate of return, 7 compounded
quarterly or 7.2 compounded semianually?
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5.3 Effective Rate and Annuities
1.071 1 .071 7.1
1.073 1 .073 7.3
7.2 compounded semiannually is better.
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5.3 Effective Rate and Annuities
What is the better rate of return, 8 compounded
monthly or 8.2 compounded quarterly?
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5.3 Effective Rate and Annuities
8.3
8.5
8.2 quarterly is better.
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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
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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

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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
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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
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5.4 Logarithmic Functions

• log 4 16 2 ? 42 16
• log 3 81 4 ? 34 81
• log10 100 2 ? 102 100

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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)
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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 gt 1
• Decreasing for 0 lt a lt 1
• Common Point (1, 0)

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5.4 Logarithmic Functions
Find the inverse of g(x) 3x.
Note The function and its inverse are
symmetrical about the line y x.
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5.4 Logarithmic Functions
Find the inverse of g(x) ex.
ln x is called the natural logarithmic function
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5.4 Logarithmic Functions
So
So
So
So
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5.4 Logarithmic Functions

1. loga(ax) x for all x ? ?
2. alog ax x for all x gt 0
3. loga(xy) logax logay
4. loga(x/y) logax logay
5. logaxn n logax

Common Logarithm log 10 x log x Natural
Logarithm log e x ln x All the above
properties hold.
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5.4 Logarithmic Functions
Product Rule
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5.4 Logarithmic Functions
Quotient Rule
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5.4 Logarithmic Functions
Power Rule
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5.4 Logarithmic Functions
Expand
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5.4 Logarithmic Functions
Find an equation of best fit for the data (1,3),
(2,12), (3,27), (4,48)
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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 gt 1
• Decreasing for 0 lt a lt 1
• Common Point (1, 0)

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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)
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5.5 Graphs of Logarithmic Functions
3 0
5 1
11 2
29 3
Sketch the graph of
Domain (2,?) Range (-?, ?)
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5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-2,?) Range (-?, ?)
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5.5 Graphs of Logarithmic Functions
Sketch the graph of
Domain (-3,?) Range (-?, ?)
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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
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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?
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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

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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

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5.6 Solving Exponential Equations

• Take the log of both sides

• Power rule

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5.6 Solving Exponential Equations

• Solve for x

• Divide

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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

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5.6 Solving Exponential Equations

• Write the left side as a single logarithm

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5.6 Solving Exponential Equations

• Equate the arguments

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5.6 Solving Exponential Equations

• Solve for x

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5.6 Solving Exponential Equations
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5.6 Solving Exponential Equations

• Check for extraneous solutions.

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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

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5.6 Solving Exponential Equations

• Write the left side as a single logarithm

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5.6 Solving Exponential Equations

• Write as an exponential equations

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5.6 Solving Exponential Equations

• Solve for x

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