Precalculus Guided Notes 3.2

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_ax 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
3⁴ = 27 so x = 4

Example 2: log ₄ 1024 = 5

Example 3: log _a 1 = 0

Properties of Logarithms

1. log _a1 = 0 because a⁰ = 1.

2. log _aa = 1 because a¹ = a.

3. log _aa^x = x and a^(log_ax) = x. 4 by inverse properties

4. If log_a x = log_ay, then x = y by One to One Property

Example: Graph of y = log_ax , 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 _ax 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 e⁰ = 1

2. ln e = 1 because e¹ = 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 ________________