Math 1014: Precalculus with Transcendentals
Ch. 4: Exponential and Logarithmic Functions
Sec. 4.2: Logarithmic Functions
I.
Logarithmic Functions
A. Definitions
b > 0, b ≠ 1, y = log b x is equivalent to b y = x . The function
f (x) = log b x is the logarithmic function with base b.
1. For x > 0 and
10 is called the common logarithmic function.
The function f (x) = log10 x is usually expressed as f (x) = log x .
2. The logarithmic function with base
e is called the natural logarithmic function. The
function f (x) = log e x is usually expressed as f (x) = ln x .
3. The logarithmic function with base
Note: A logarithmic function is the inverse of an exponential function.
B. Location of Base and Exponent in Exponential and Logarithmic Forms
Logarithmic Form:
y = log b x
Exponential Form:
by = x
⎫
⎪
⎬ b is the base and y is the exponent
⎪
⎭
C. Examples
1. Write each equation in its equivalent exponential form:
a.
4 = log 2 16
b.
5 = log b 32
2. Write each equation in its equivalent logarithmic form:
1
125
a.
5 −3 =
b.
3
c.
b 3 = 343
64 = 4
II.
Basic Logarithmic Properties
A. Basic Properties Involving One
1.
log b b = 1 because b1 = b
2.
log b 1 = 0 because b 0 = 1
B. Inverse Properties of Logarithms
For
b > 0 and b ≠ 1 ,
1.
log b b x = x
2.
b logb x = x
C. General Properties
Common Log Properties
Natural Log Properties
log b b = 1
log10 = 1
ln e = 1
log b 1 = 0
log1 = 0
ln1 = 0
log b b x = x
log10 x = x
ln e x = x
b logb x = x
10 log x = x
eln x = x
D. Examples
1. Evaluate/simplify without a calculator
a.
log 3 27
h.
log 1000
b.
log 3
i.
log 10 8
c.
log 6 6
j.
10 log
d.
log 5 5
k.
ln e
e.
log 6 1
l.
ln e6
f.
log 4 4 6
m.
ln
g.
7 log 7 23
n.
eln 300
o.
ln e13x
1
9
53
1
e7
III. Graphs of Logarithmic Functions
A. Domain and Range for
y = log b x
{x | x > 0} or (0,∞)
2. Range: R or (−∞,∞)
1. Domain:
3. Examples
Find the domain of the following:
a.
f (x) = log 4 x
Domain:
b.
f (x) = log (7 − x)
Domain:
c.
f (x) = log (x − 7)2
Domain:
NOTE: In general, the domain of
B. Graphs of
1.
f (x) = log b (g(x)) consists of all x for which g(x) > 0 .
y = log b x and y = b x
b >1
2.
0 < b <1
y
y
y = bx , b > 1
y = bx , 0 < b < 1
y=x
y=x
1
1
f (x) = log b x , b > 1
x
1
1
x
f (x) = log b x , 0 < b < 1
C. Characteristics of Logarithmic Functions of the Form
1. The domain of
of
f (x) = log b x
f (x) = log b x consists of all positive real numbers: (0,∞) . The range
f (x) = log b x consists of all real numbers: (−∞,∞) .
f (x) = log b x pass through the
point (1,0) because f (1) = log b 1 = 0 . The x-intercept is 1. There is no y-intercept.
2. The graphs of all logarithmic functions of the form
3. If
b > 1, f (x) = log b x has a graph that goes up to the right and is an increasing
function.
4. If
0 < b < 1, f (x) = log b x has a graph that goes down to the right and is a decreasing
function.
5. The graph of
f (x) = log b x approaches, but does not touch the y-axis. The y-axis, or
x=0, is a vertical asymptote.
D. Transformations of Logarithmic Functions
In each case,
Transformation
c represents a positive real number.
Equation
Description
Vertical
g(x) = log b x + c
Shifts the graph of
f (x) = log b x upward c units
translation
g(x) = log b x − c
Shifts the graph of
f (x) = log b x downward c units
Horizontal
g(x) = log b (x + c) Shifts the graph of f (x) = log b x to the left c units
translation
g(x) = log b (x − c) Shifts the graph of f (x) = log b x to the right c units
Reflection
g(x) = − log b x
Reflects the graph of
f (x) = log b x about the x-axis
g(x) = log b (−x)
Reflects the graph of
f (x) = log b x about the y-axis
Vertical
stretching
g(x) = c log b x
or shrinking
Horizontal
stretching
or shrinking
g(x) = log b (cx)
Vertically stretches the graph of
Vertically shrinks the graph of
f (x) = log b x if c > 1
f (x) = log b x if 0 < c < 1
Horizontally shrinks the graph of
f (x) = log b x if c > 1
Horizontally stretches the graph of
f (x) = log b x if 0 < c < 1