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```Exponential and Logarithmic Functions & their Graphs
Exponential Functions and Graphs
Definition of an Exponential Function:
The exponential function
where
, and
with base
is defined by
For all real numbers , the function defined by
is any real number.
is called the natural exponential function.
Notice that the base of the exponential function is required to be positive and cannot be equal to 1.
Properties of an exponential function:
For all positive real numbers
, the exponential function defined by
has the following properties:
1.
has the set of real numbers as its domain.
5.
is a one-to-one function.
2.
has the set of positive real numbers as its range.
6.
is an increasing function if
3.
has a graph with -intercept of
7.
is a decreasing function if
4.
has a graph asymptotic to the -axis.
axis.
.
Figure A
. See Figure A.
. See Figure B.
Figure B
.
.
(0,1)
(0,1)
Logarithmic Functions and Graphs
Definition of Logarithmic Function:
The logarithmic function
where
is a positive constant,
with base
, and
is defined by
is any positive real number.
Note: Logarithmic functions are inverses of the corresponding exponential functions.
Properties of a logarithmic function:
For all positive real numbers
, the function defined by
has the following properties:
1.
has the set of positive real numbers as its domain.
5.
is a one-to-one function.
2.
has the set of real numbers as its range.
6.
is an increasing function if
3.
has a graph with an -intercept of
7.
is a decreasing function if
4.
has a graph asymptotic to the -axis.
axis.
Figure C
.
(1,0)
.
Figure D
.
(1,0)
. See Figure C.
. See Figure D.
Practice Problems
Sketch graphs for each of the following exponential and logarithmic functions and label the
intercepts:
1. 4
8. 2 log 2. 1 9. log 4
3. ln 1
10. log 1
4. log 11. ln 3
5. log 12. ln 2
6. log 13. 5 ln 7. 2 log 14. 3 ln Practice Problems Answers
1.
4.
2.
5.
3.
6.
7.
11.
8.
12.
9.
13.
10.
14.
```