Precalculus 3.2 Logarithmic Functions and their Graphs
Recall the exponential function f (x) = ax, 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 ax if and only if x = ay.
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 = 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) = 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 e0 = 1
2. ln e = 1 because e1 = e.
3. ln ex = 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 ___________________
