Precalculus 3.2 Logarithmic Functions and their Graphs
Recall the exponential function f (x) = ax, 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 = logax if and only if x = ay.
The function f (x) = log a x is called the logarithmic function with base a.
Examples 1: log 3 81 =
so we think: 3x = 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 aax = x and a(logax) = x. 4 by inverse properties
4. If loga x = logay, then x = y by One to One Property
Example: Graph of y = logax , a > 1
Domain: (0, ∞ )
Range: (- ∞, ∞ )
Intercept: ( 1, 0)
y-axis is a vertical asymptote
(log a x ) → - ∞ as x → 0+
Reflection of graph of y = ax 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) = loga 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 ex = x and e ln x = x by inverse properties.
4. If ln x = ln y, then x = y by One - to - one properties
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 ________________