Precalculus 3.2 Logarithmic Functions and their Graphs

*Recall the exponential function f* (*x*) = a^{x}, a > 0, a

1 . Every function of

this form passes the Horizontal Line Test and therefore must have an inverse. This

inverse function is called the *logarithmic function *with base *a*.

Definition of Logarithmic Function:

For *x* > 0 and 0 < *a*

1,

*y* = log * _{a}*x if and only if

*x*=

*a*

^{y}.

The function

*f *(*x*) = log_{ a }*x*

is called the *logarithmic function *with base *a*.

Examples:

log _{3} 27 = ____________ log _{4} 64= _________ log _{a} 1 = ___________

Properties of Logarithms

1. log _{a} 1 = ________ because *a*^{ 0} = 1

2. log _{a} *a* = ________ because *a*^{ 1} = *a*.

3. log_{ a }*a ^{x}* = _________because

*a*^(log

_{a}

*x*) =

*x*

4. If log _{a} *x* = log _{a} *y*, then *x* = *y*.

Example:

Graph of *y* = log _{a} *x* , *a* > 1

Domain: (0, ∞ )

Range: (- ∞, ∞ )

Intercept: ( 1, 0)

Increasing

*y*-axis is a vertical asymptote

(log_{ a} *x* ) → - ∞ as *x* → 0^{+}

Continuous

Reflection of graph of *y* = *a*^{x} in the line *y* = *x*.

Transformations of Graphs of Logarithmic Functions:

a. *f*(*x*) = log _{a} (*x* - h) + k Basic graph - h = ________ k = ________ a = __________

b. *f* (*x*) = log _{a} (*x* - 2) This is the graph *f* (*x*) = log _{a} *x* shifted ___ units to the

________.

c. *f* (*x*) = log _{a} (*x* +3) This is the graph *f* (*x*) = log _{a} *x* shifted ___ units to the

________.

d. *f* (*x*) = log _{a} *x* +5This is the graph *f* (*x*) = log _{a} *x* shifted ___

units________.

e. *f* (*x*) = log _{a} (-*x*) This is the graph *f* (*x*) = log _{a} *x* reflected in the ______.

f. . *f* (*x*) = -log _{a} x This is the graph *f* (*x*) = log _{a} *x* reflected in the ______.

Examples:

The Natural Logarithmic Function

The function defined by

*f* (*x*) = log _{e} *x* = ln *x*, *x* > 0

is called the natural logarithmic function.

Properties of Natural Logarithms:

1. ln 1 = 0 because e^{0} = 1

2. ln e = 1 because e^{1} = e.

3. ln e^{x} = *x* and e ^{ln x }= *x*.

4. If ln *x* = ln *y*, then *x* = *y*.

Example:

1. Find the domain, the intervals in which the function is increasing or decreasing,

and approximate any relative maximums or minimums:

*f* (*x*) = *x*/(ln *x*)

a. Graph the function:

1. Domain = ___________

2. Increasing = ____________

3. Decreasing = ____________

4. Maximum or minimums ___________________