lesson 3-2 logarithmic functions and their graphs
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Lesson 3-2
Logarithmic Functions
And
Their Graphs
Logarithmic Functions and Their Graphs
When one looks at the graph of the exponential function y = ax, by applying the horizontal line test, one can seethat this has an inverse function. This inverse function iscalled a logarithm. The sole function of a logarithm is tofind an exponent!
Read the history of logarithms found at:http://www.spiritus-temporis.com/logarithm/history.html
Logarithmic Functions and Their Graphs
Definition of Logarithmic Function
For x > 0, a > 0, and a 1,
y = loga x if and only if x = a y.
The function given by
f(x) = loga x
is called the logarithmic function with base a.
Logarithmic Functions and Their Graphs
Examples:
1. Evaluate f(x) = log4x when x = 64.
Solution: f(64) = log464 which means 4 to what powerequals 64? 43 = 64
Therefore, f(64) = log464 = 3 (Remember that the purpose of a logarithm is to find anexponent!)
Logarithmic Functions and Their Graphs
5
1 1 1log which means5 towhat power gives ?
25 25 25f
5
1 1Therefore, log 2
25 25f
Example 2:
Evaluate f(x) = log5x when x = 1/25.
Solution:
1
12 2
1Remember that may be rewritten as 25 which
25
then leads to 5 5 .or
Logarithmic Functions and Their Graphs
21 log 1 which means2 towhat power gives1?f
0Remember 1.a
Exampe 3:
Evaluate f(x) = log2x when x =1.
Solution:
2Therefore, 1 log 1 0f
Logarithmic Functions and Their Graphs
Try the following:
1. f(x) = log6x when x = 216
2. f(x) = log4x when x = 1
3. f(x) = log4x when x = 1/1024
Solutions:
f(216) = log6216 = 3
f(1) = log41 = 0
f(1/1024) = log4(1/1024) = - 5
Logarithmic Functions and Their Graphs
Some Basic Properties of Logarithms:
0
1
1. log 1 0 1.
2. log 1 .
3. log .
4. If log log , then .
a
a
x x xa
a a
because a
a because a a
a x because a a
x y x y
Logarithmic Functions and Their Graphs
Now let’s look at the graph of a logarithmic function.
Remember that when a graph has an inverse, the x- andy-coordinates change positions. Look at the followingtables.
y = 3x
x y-3 1/27-2 1/9-1 1/30 11 32 93 27
y = log3x x y1/27 -31/9 -21/3 -11 03 19 227 3
Its inverse is
Inverse function
Logarithmic Functions and Their Graphs
Graphing y = log3x
Logarithmic Functions and Their Graphs
Now you graph y = log2x
Logarithmic Functions and Their Graphs
,
1, 0
Now let’s compare both graphs:
What is the domain for each graph?
What is the range for each graph?
What is the x-intercept for each graph?
What appears to be the vertical asymptote for each graph?
Is the function increasing or decreasing? If so, what is the interval?
0x
increasing, the interval is 0,
0,
Logarithmic Functions and Their Graphs
logay x
4log 3y x
6log 5y x
What would the function be with the following transformations?
1. The reflection of y = logax?
2. A horizontal shift of 3 units to the left for y = log4x?
3. A vertical shift of 5 units up for y = log6x?
4. The reflection of y = logax with a horizontal shift of c units and a verical shift of d units? logay x c d
Logarithmic Functions and Their Graphs
The Natural Logarithmic Function
If y = ex is the natural exponential function, then y = ln x is the natural logarithmic function.
For x > 0,
y = ln x if and only if x = ey.
The function is given by
f(x) = logex = ln x
is called the natural logarithmic function.
Logarithmic Functions and Their Graphs
Graph the natural exponential function, the naturallogarithmic function, and the line of symmetry all onone graph.
Logarithmic Functions and Their Graphs
Properties of Natural Logarithms
1. ln 1 = 0 Why? Because e 0 = 1
2. ln e = 1 Why? Because e 1 = e
3. ln ex = x Why?Because in exponential formif the two bases are the same,then the exponents must beequal.
4. If ln x = ln y, then x = y Why?
Because since the two bases are the same, the tworesults must be equal.
Logarithmic Functions and Their Graphs
What you should know:
1. What a logarithm is
2. Basic properties for logarithms
3. How to graph a logarithmic function
4. Transformations to a logarithmic function
5. The natural logarithmic function and its basic properties