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