**Precalculus 3.2 Logarithmic Functions and their Graphs**

Recall the exponential function f (x) = a

^{x}, a > 0, a is not equal to 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 is less than a which is not equal to 1, then

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 1:**log

_{3}81 =

so we think: 3

^{x}= 81

3

^{4}= 27 so x = 4

**Example 2**: log

_{4}1024 = 5

**Example 3**: log

_{a}1 = 0

**Properties of Logarithms**

1. log

_{a}1 = 0 because a

^{0}= 1.

2. log

_{a}a = 1 because a

^{1}= a.

3. log

_{a}a

^{x}= x and a

^{(logax)}= x. 4 by inverse properties

4. If log

_{a}x = log

_{a}y, then x = y by One to One Property

**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 2 units to the right.

c. f (x) = log

_{a}(x +3) This is the graph

f (x) = log

_{a}x shifted 3 units to the left.

d. f (x) = log

_{a}x +5 This is the graph

f (x) = log

_{a}x shifted 5 units up.

e. f (x) = log

_{a}(-x) This is the graph

f (x) = log

_{a}x reflected in the y-axis.

f. . f (x) = -log

_{a}x This is the graph

f (x) = log

_{a}x reflected in the x-axis.

**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 by inverse properties.

4. If ln x = ln y, then x = y by One - to - one properties

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 ________________