Logarithmic Functions
Definition of a Logarithmic Function
• For x > 0 and b > 0, b = 1,
• y = logb x is equivalent to by = x.
• The function f (x) = logb x is the
logarithmic function with base b.
Location of Base and Exponent in
Exponential and Logarithmic Forms
Exponent
Exponent
Logarithmic form: y = logb x
Base
Exponential Form: by = x.
Base
Text Example
Write each equation in its equivalent exponential form.
a. 2 = log5 x
b. 3 = logb 64
c. log3 7 = y
Solution With the fact that y = logb x means by = x,
a. 2 = log5 x means 52 = x.
b. 3 = logb 64 means b3 = 64.
Logarithms are exponents.
c. log3 7 = y or y = log3 7 means 3y = 7.
Logarithms are exponents.
Text Example
Evaluate
a. log2 16
b. log3 9
c. log25 5
Solution
Logarithmic
Expression
Question Needed for
Evaluation
Logarithmic Expression
Evaluated
a. log2 16
2 to what power is 16?
log2 16 = 4 because 24 = 16.
b. log3 9
3 to what power is 9?
log3 9 = 2 because 32 = 9.
c. log25 5
25 to what power is 5?
log25 5 = 1/2 because 251/2 = 5.
Basic Logarithmic Properties
Involving One
• Logb b = 1
because 1 is the exponent to
which b must be raised to obtain b. (b1 = b).
• Logb 1 = 0
because 0 is the exponent to
which b must be raised to obtain 1. (b0 = 1).
Inverse Properties of Logarithms
For x > 0 and b  1,
logb bx = x
The logarithm with base b of
b raised to a power equals that power.
b logb x = x
b raised to the logarithm with
base b of a number equals that number.
Text Example
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution We first set up a table of coordinates for f (x) = 2x. Reversing these
coordinates gives the coordinates for the inverse function, g(x) = log2 x.
x
-2
-1
0
1
2
3
x
1/4
1/2
1
2
4
8
f (x) = 2x
1/4
1/2
1
2
4
8
g(x) = log2 x
-2
-1
0
1
2
3
Reverse coordinates.
Text Example cont.
Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.
Solution
We now plot the ordered pairs in both tables, connecting them with smooth
curves. The graph of the inverse can also be drawn by reflecting the graph
of f (x) = 2x over the line y = x.
y=x
f (x) = 2x
6
5
4
3
f (x) = log2 x
2
-2
-1
-1
-2
2
3 4
5
6
Characteristics of the Graphs of Logarithmic
Functions of the Form f(x) = logbx
• The x-intercept is 1. There is no y-intercept.
• The y-axis is a vertical asymptote.
• If b > 1, the function is increasing. If 0 < b < 1,
the function is decreasing.
• The graph is smooth and continuous. It has no
sharp corners or edges.
Properties of Common Logarithms
General Properties
1. logb 1 = 0
2. logb b = 1
3. logb bx = x
4. b logb x = x
Common Logarithms
1. log 1 = 0
2. log 10 = 1
3. log 10x = x
4. 10 log x = x
Examples of Logarithmic
Properties
log b b = 1
log b 1 = 0
log 4 4 = 1
log 8 1 = 0
3 log 3 6 = 6
log 5 5 3 = 3
2 log 2 7 = 7
Properties of Natural Logarithms
General Properties
1. logb 1 = 0
2. logb b = 1
3. logb bx = x
4. b logb x = x
Natural Logarithms
1. ln 1 = 0
2. ln e = 1
3. ln ex = x
4. e ln x = x
Examples of Natural Logarithmic
Properties
log e e = 1
log e 1 = 0
e log e 6 = 6
log e e 3 = 3
Logarithmic Functions