LESSON 5: DEFINITION AND LAWS OF LOGARITHMS
From WikiEducator
Objectives

Contents
Definition
The log of a number is the power to which the base must be raised to give that number.
Laws of logarithms
 The multiplication law
 The division law
 The power law
The multiplication law
Let
log_{b}M = x
and
log_{b}N = y
or in index form
M = b^{x}
and
N =b^{y}
Now
MN =b^{x} x b^{y}
and
MN =b^{x+y}
or in log form
log_{b}MN = x + yHence
log_{b}MN = log_{b}M + log_{b}N 
The division law
Now
M/N = b^{x}/b^{y}
and
M/N = b^{(xy)}
or in log form
log_{b}M/N = xy
Hence
log_{b}M/N =log_{b}M  log_{b}N 
The power law
Now
M^{n} = (b^{x})^{n}
and
M^{n} = b^{nx}
or in log form
log_{b}M^{n} = n(log_{b}M) 
Other special logs
The value of log_{b}1
Let
log_{b}1 = x
then in index form
1 =b^{x}
Hence
log_{b}1 = 0
therefore
To any base the value of log1 is zero 
The value of log_{b}b
Let
log_{b}b = x
then in index form
b = b^{x}
Hence
log_{b}b = 1
Therefore
The value of the log of a number to the same base is unity 
The value of log_{b}0
Let
log_{b}o = x
then in index form
0 = b^{x}
Hence
log_{b}0 = infinity
Therefore
To any base the log of zero is minus infinity 
The value of log_{b}(N)
Let
log_{b}(N) = x
then in index form
N = b^{x}
Hence
log_{b}(N) has no real value
Therefore
Only positve numbers have real logarithms 
Worked examples
question
Find the value of x in each of the following
a. log_{x}9 = 2
solution
in index form
x^{2} = 9
and
x =√9
therefore
x = 3 ans
b. log_{7}x = 0
solution
in index form
x = 7^{0}
therefore
x = 1 ans
Malackt 15:49, 26 February 2007 (CET)