CRP 762 Handout
Logarithm
Source: http://www.sosmath.com/algebra/algebra.html (Section: Logarithms and Exponential
Functions)
Definition of Exponential Function
The exponential function f with base a is denoted by f ( x)  a x , where a  1 , and x is any real
number. The function value will be positive because a positive base raised to any power is
positive. This means that the graph of the exponential function f ( x)  a x will be located in
quadrants I and II.
For example, if the base is 2 and x = 4, the function value f(4) will equal 16. A corresponding
point on the graph of f ( x)  2 x would be (4, 16).
1
Definition of Logarithmic Function
For x >0, a>0 , and a  1 , we have
Since x > 0, the graph of the above function will be in quadrants I and IV.
Comments on Logarithmic Functions




The exponential equation 4 3  64 could be written in terms of a logarithmic equation as
log 4 (64)  3 .
1
The exponential equation 5  2 
can be written as the logarithmic equation
25
1
log 5 ( )  2 .
25
Since logarithms are nothing more than exponents, you can use the rules of exponents
with logarithms.
Logarithmic functions are the inverse of exponential functions. For example if (4, 16) is a
point on the graph of an exponential function, then (16, 4) would be the corresponding
point on the graph of the inverse logarithmic function.
2

The two most common logarithms are called common logarithms and natural
logarithms. Common logarithms have a base of 10, and natural logarithms have a base of
e. On your calculator, the base 10 logarithm is noted by log, and the base e logarithm is
noted by ln.
3
Properties of Logarithms
Property 1: log a 1  0 because
.
Example 1: In the equation 14 0  1 , the base is 14 and the exponent is 0. Remember that
a logarithm is an exponent, and the corresponding logarithmic equation is
log 14 1  0 where the 0 is the exponent.
1
1
Example 2: In the equation ( ) 0  1 , the base is and the exponent is 0. Remember that
2
2
a logarithm is an exponent, and the corresponding logarithmic equation is log 1 1  0 .
2
Example 3: Use the exponential equation x 0  1 to write a logarithmic equation. The
base x is greater than 0 and the exponent is 0. The corresponding logarithmic equation is
log x 1  0 .
Property 2: log a a  1 because
.
Example 4: In the equation 31  3 , the base is 3, the exponent is 1, and the answer is 3.
Remember that a logarithm is an exponent, and the corresponding logarithmic equation is
log 3 3  1 .
Example 5: In the equation 871  87 , the base is 87, the exponent is 1, and the answer is
87. Remember that a logarithm is an exponent, and the corresponding logarithmic
equation is log 87 87  1 .
Example 6: Use the exponential equation p1  p to write a logarithmic equation. If the
base p is greater than 0, then log p p  1 .
Property 3: log a a x  x because a x  a x .
Example 7: Since you know that 3 4  3 4 , you can write the logarithmic equation with
base 3 as log 3 3 4  4 .
Example 8: Since you know that 13 4  13 4 , you can write the logarithmic equation with
base 13 as log 13 13 4  4 .
Example 9: Use the exponential equation 4 2  16 to write a logarithmic equation with
base 4. You can convert the exponential equation 4 2  16 to the logarithmic equation
log 4 16  2 . Since the 16 can be written as 4 2 , the equation log 4 16  2 can be written
log 4 4 2  2 .
4
Rules of Logarithms
Let a be a positive number such that a does not equal 1, let n be a real number, and let u and v
be positive real numbers.
Logarithmic Rule 1: Log a (uv)  Log a (u)  Log a (v)
u
Logarithmic Rule 2: Log a ( )  Log a (u )  Log a (v)
v
Logarithmic Rule 3: Log a (u ) n  nLog a (u )
Since logarithms are nothing more than exponents, these rules come from the rules of exponents.
Let a be greater than 0 and not equal to 1, and let n and m be real numbers.
Exponential Rule 1: a n a m  a n  m
Example: Let a = 5, n = 2, and m = 6. 5 2 * 5 6  25 *15625  390625 and 5 26  58  390625
an
Exponential Rule 2: m  a n  m
a
Example: Let a = 5, n = 2, and m = 6.
52
25

 0.0016 and 5 26  5 4  0.0016
6
15625
5
Exponential Rule 3: (a n ) m  a nm
Example: Let a = 5, n = 2, and m = 6. (5 2 ) 6  (25) 6  244140625 and
(5) 2*6  (5)12  244140625
5
Note: The above information is from http://www.sosmath.com/. This is good website for a quick
catch of the Math you have forgotten. It has lots of examples. The only thing I don’t like about
this website is that when you open it, it has some pop-ads.
Here you can select the section you want. Algebra and Matrix
Algebra may be the materials we need in our class.
6