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Exponential FunctionsExponential Functions
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Exponential FunctionsExponential Functions
Def: An exponential function with base a is a function of the form f(x) = ax , where a and x are real numbers such that a > 0 and a ≠ 1.
i.e. The base is constant and the exponent is a variable.
Properties of Exponential Functions
(1) f(x) is increasing for a > 1 and decreasing for 0 < a < 1.
(2) The y-intercept is at (0,1).
(3) The x-axis is a horizontal asymptote.
(4) Has domain (-∞, ∞) and range (0, ∞).
(5) f(x) is a one-to-one function.
Exponential Family of Functions (Graphing)
y = abx+c + d (1)By Table…(pick a number
according to properties listed above, use a calculator if necessary)
(2) By Transformation (translating, stretching, shrinking and reflecting)
• Example: the graph of y = 2x
• Notice the y-intercept is at (0,1)
• The x-axis is a horizontal Asymptote, i.e. the line which the graph of y = 2x approaches as x→ -
but never crosses.
Graphing
∞
• Graph of
Graphing
x
y
2
1
x
y
2
1
• • Translate (i.e. shifted)
3 units to the left• The domain is
(-∞,∞).
Graphing
32 xy
32 xy
Logarithmic FunctionsLogarithmic Functions
• Def: For a > 0 and a ≠ 1 , the logarithmic function with base a is denoted
• f(x) = loga(x), where y = loga(x) if and only if ay = x .
• i.e. instead of f -1(x) we use loga(x) as the inverse of an exponential function
Two Popular BasesTwo Popular Bases
(1) Common log…base 10 (log)(2) Natural log…base e (ln)
Properties of Logarithmic Functions1) f(x) is increasing for a >1 and decreasing for
0<a< 12) The x-intercept is at (1,0)3) The y-axis is a vertical asymptote4) Has Domain (0,∞) and Range (-∞,∞)5) f(x) is a one-to-one function
Logarithmic Family of Functions Logarithmic Family of Functions (Graphing)(Graphing)
Graphing Logarithms
1.By Setting up table
2.By Transformation
• The common Log is base 10, when the base a is not written, it is understood to be base 10.
• The x-intercept (i.e. zero or root) is (1,0)• The vertical asymptote is y-axis
ExampleExample
y = log (x)
Graph of Logarithmic FunctionsGraph of Logarithmic Functions
xy log
• The exponential function ex and Natural Log (Ln) are inverses of each other, notice the reflection over the diagonal line y = x
• Note: e ≈ 2.71828…• Note: Ln rises more rapidly
than the common log
v.s
Exponential Functions vs Logarithmic Functions
xe xln
• Shifted one points to the right, (x-1)
• New vertical asymptote at x = 1
• Zero at (2, 0)
• Domain is from (-∞,∞)
ExampleExample
y = log (x – 1)
Graph of a Logarithmic Graph of a Logarithmic TransformationTransformation
y = log (x - 1)
Exponential and Logarithmic Exponential and Logarithmic Functions LinksFunctions Links
• Exponential Functions Handout
• Algebra and Logarithmic Functions Handout
• Exponential Functions Quiz