Transcript Logarithmic Functions Definition of a Logarithmic Function For x > 0 and b > 0, b = 1, y = logb x.

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
raised to obtain b.
• Logb 1 = 0
raised to obtain 1.
because 1 is the exponent to which b must be
(b1 = b).
because 0 is the exponent to which b must be
(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.
Properties of Common Logarithms
General Properties
1. logb 1 = 0
2. logb b = 1
3. logb bx = 0
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 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 = 0
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
e log e 6 = e ln 6 = 6
log e e 3 = 3
Problems
Use the inverse properties to simplify:
1. ln e
3. e
7x
ln x
5. log107.1
2. e
ln 4 x 2
4. log1000
6. 10log e
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. (x = 0)
• If 0 < b < 1, the function is decreasing. If b > 1, the function is
increasing.
• The graph is smooth and continuous. It has no sharp corners or
edges.
6
5
6
5
4
4
3
3
2
-2
-1
-1
-2
2
3 4
5
6
f (x) = logb x
0<b<1
2
-2
-1
-1
-2
2
3 4
5
6
f (x) = logb x
b>1
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
Graph f (x) = 2x and g(x) = log2 x in the same rectangular
coordinate system.
Solution
We now sketch the basic exponential graph. The graph of the inverse
(logarithmic) 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
Examples
Graph using transformations.
f ( x)  log( x  1)
g ( x)   log( x  1)
h( x)  1  ln x
Domain of Logarithmic Functions
Because the logarithmic function is the inverse of the exponential
function, its domain and range are the reversed.
The domain is { x | x > 0 } and the range
all real numbers.
f ( x) will
logbe
b  x  c
For variations of the basic graph, say
the domain
will consist of all x for which x + c > 0.
Find the domain of the following:
1.
h( x)  log4  x  5
2.
g ( x)  log  x  3
3.
f ( x)  ln  x  3
2
Sample Problems
Find the domain of
f ( x)  log 4 ( x  3)
g ( x)  ln(2 x  1)
h( x)  log(4  x)