Exponential Functions

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Exponential Functions Logarithmic Functions
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Definition of the Exponential Function
The exponential function f with base b is defined
by f (x) bx or y bx Where b is a positive
constant other than and x is any real number.
Here are some examples of exponential
functions. f (x) 2x g(x) 10x h(x) 3x1
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Characteristics of Exponential Functions

• The domain of f (x) bx consists of all real
  numbers. The range of f (x) bx consists of all
  positive real numbers.
• The graphs of all exponential functions pass
  through the point (0, 1) because f (0) b0 1.
• If b gt 1, f (x) bx has a graph that goes up to
  the right and is an increasing function.
• If 0 lt b lt 1, f (x) bx has a graph that goes
  down to the right and is a decreasing function.
• f (x) bx is a one-to-one function and has an
  inverse that is a function.
• The graph of f (x) bx approaches but does not
  cross the x-axis. The x-axis is a horizontal
  asymptote.

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Transformations Involving Exponential Functions
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Example
Use the graph of f (x) 3x to obtain the graph
of g(x) 3 x1.
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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 2.72. More accurately,
The number e is called the natural base. The
function f (x) ex is called the natural
exponential function.
-1
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Formulas for Compound Interest

• After t years, the balance, A, in an account
  with principal P and annual interest rate r (in
  decimal form) is given by the following formulas
• For n compounding per year
• For continuous compounding A Pert.

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Example

• Use A Pert to solve the following problem Find
  the accumulated value of an investment of 2000
  for 8 years at an interest rate of 7 if the
  money is compounded continuously
• Solution
• A PertA 2000e(.07)(8)A 2000 e(.56)A
  2000 1.75A 3500

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Exponential Functions

• Exponential Function An equation in the form
  f(x) Cax.
• Recall that if 0 lt a lt 1 , the graph
  represents exponential decay
• and that if a gt 1, the graph represents
  exponential growth
• Examples f(x) (1/2)x f(x) 2x

Exponential Decay
Exponential Growth
We will take a look at how these graphs shift
according to changes in their equation...
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How do the following graphs compare to the
original graph of f(x) (1/2)x ?

• f(x) (1/2)x f(x) (1/2)x 1
  f(x) (1/2)x 3

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How do the following graphs compare to the
original graph of f(x) (2)x ?

• f(x) (2)x f(x) (2)x 3
  f(x) (2)x 2 3

(3,1)
(0,1)
(-2,-2)
Notice that f(0) 1
Notice that this graph is shifted 3 units to
the right.
Notice that this graph is shifted 2 units to
the left and 3 units down.
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• The Logarithmic Function
• The logarithm function to the base a, where a gt 0
  and a ? 1, is denoted by
• f (x) log a x or y log a x
• (read as f(x)or y is the logarithm to the base
  a of x) and is defined by
• f (x) log a x if and only if x a y
• where the domain is xgt0 and the range is -ltylt.

log a x is the exponent to which the base a must
be raised to give x
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• Graphing a Logarithmic Functions f (x) log a
  x
• The domain is the set of positive real numbers
  the range is all real numbers.
• The x-intercept of the graph is 1. There is no
  y-intercept.
• The y-axis (x 0) is a vertical asymptote of the
  graph.
• A logarithmic function is decreasing if 0 lt a lt 1
  and increasing if a gt 1
• The graph of f contains the points (1, 0) and (a,
  1).
• The graph is smooth and continuous, with no
  corners or gaps.

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Characteristics of the Graphs of Logarithmic
Functions of the form f(x) log b x

• The x-intercept is 1. There is no y-intercept.
• The y-axis is a vertical asymptote. (x 0)
• If 0 lt b lt 1, the function is decreasing. If b gt
  1, the function is increasing.
• The graph is smooth and continuous. It has no
  sharp corners or edges.

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Logarithmic Graph
Exponential Graph
Graphs of inverse functions are reflected about
the line y x
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Domain of Logarithmic Functions

• Because the logarithmic function is the inverse
  of the exponential function, its domain and range
  are the reversed.
• The domain is x x gt 0 and the range will be
  all real numbers.
• For variations of the basic graph, say
  the domain will consist of all x for
  which x c gt 0.
• Find the domain of the following
• 1.
• 2.
• 3.

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Properties of Logarithms