Logarithm
Logarithms were developed to solve the
complicated problems in very simple and easy way. In Science, we need to
calculate the 8^{th} or 10^{th} (or higher) power of decimal
numbers. Suppose we want to calculate (4.302)^{8}, then its calculation
is very complex. Similarly, if you have to find the 5^{th} power of
2.3521, the calculation of this problem will be very complex. Logarithm is written
as log in short form.
Let us assume a problem.
Find the volume of a sphere whose radius
is 6.235 cm.
The volume of a sphere (V) = 4/3 πr^{3}
= 4/3 (3.14) × (6.235)^{3}
The calculation of this problem is also
very complex. To calculate this type of sums, we use logarithms.
Logarithm Definition
For a given positive real number M and a positive
real number a ≠ 1, if there exists a unique real number x such
that a^{x} = M, then x is defined to be logarithm
of the number M to the base a. Thus, logarithm of a positive number is
defined and it is written as log_{a }M = x.
For example, 2^{3} = 8, then power ‘3’ is called the logarithm of the number 8 to the base 2, and written as log_{2 }8 = 3.
It is to be noted that a^{x }= M
is called the exponential form and log_{a} M = x is
called the logarithmic form.
The following examples will explain clearly about Exponential form and Logarithm form.
Exponential Form |
Logarithm Form |
2^{6} = 64 |
log_{2 }64 = 6 |
3^{3} = 27 |
log_{3 }27 = 3 |
4^{4} = 256 |
log_{4 }256 = 4 |
10^{0} = 1 |
log_{10 }1 = 0 |
10^{1} = 10 |
log_{10 }10 = 1 |
10^{2} = 100 |
log_{10 }100 = 2 |
10^{-1} = 0.1 |
log_{10 }0.1 = -1 |
10^{-2} = 0.01 |
log_{10 }0.01 = -2 |
Logarithm Rules or Laws of Logarithm
1. Product Rule:
log_{a} (MN) = log_{a} M + log_{a}
N
2. Quotient Rule:
log_{a} (M/N) = log_{a} M - log_{a}
N
3. Power Rule:
log_{a} (M^{N}) = N.log_{a}
M
4. Zero Rule:
The logarithm of 1 to any base is zero, i.e., log_{x}
1 = 0.
5. Identity Rule:
The
logarithm of a number to the same base is equal 1,
i.e., log_{x}
x = 1.
6. Inverse Property of Logarithm:
log_{a} (a^{k}) = k.
7. Inverse Property of Exponent:
a^{loga(k)}
= k.
8. Change of Base Formula:
log_{a} x = log_{c}
x/log_{c} a
9. Base 10 may be mentioned or may not. Hence_{,}
when no base is mentioned, it is understood that the base is 10.
10. Logarithms to
the base 10 are called common logarithms.
Derivation of Logarithm Rules
Product Rule Derivation:
The logarithm of the product is the
sum of the logarithms of the factors, that is, log_{a} MN = log_{a}
M + log_{a} N
Let log_{a} M = x and log_{a}
N = y.
Changing each of these logarithms into
exponential form, we get
a^{x} = M ……… (i)
a^{y} = N ……… (ii)
Multiplying equations (i) and (ii), we
get
MN = a^{x}. a^{y}
MN = a^{x }^{+ y} (By laws of exponents)
Again, changing it back to logarithmic
form, we get
log_{a} MN = x +
y
Putting the values of x and y, we get
log_{a} MN = log_{a} M + log_{a }N
Hence proved.
Quotient Rule Derivation:
The logarithm of the ratio of the two
quantities is the logarithm of the numerator minus the logarithm of the
denominator, that is,
log_{a} M/N = log_{a}
M - log_{a} N
Let log_{a} M = x and log_{a}
N = y.
Changing each of these logarithms into
exponential form, we get
a^{x} = M ……… (i)
a^{y} = N ……… (ii)
Dividing equation (i) by (ii), we get
M/N = a^{x}/ a^{y}
M/N = a^{x }^{- y} (By laws of exponents)
Again, changing it back to logarithmic
form, we get
log_{a} M/N = x -
y
Putting the values of x and y, we get
log_{a} M/N = log_{a} M - log_{a }N
Hence proved.
Power Rule Derivation:
The logarithm of an exponential number
is the product of the exponent and the logarithm of the base, that is, log_{a}
(M^{N}) = N.log_{a} M
Let log_{a} M = x
Changing this into exponential form,
we get
a^{x} = M
Raising both sides to the power N, we
get
(a^{x})^{N} = (M)^{N}
a^{Nx}
= (M)^{N }(Using
power of power law of exponents)
Changing this back into logarithmic
form, we get
log_{a} (M)^{N} = Nx
Putting x = log_{a} M back
here, we get
log_{a} (M)^{N} = N.log_{a}
M Hence proved.
Similarly, you can solve the other
rules of the lagarithms.
Logarithm Table
Solved Examples on Logarithm Rules
Example 1: Find the value of log 25 + log 4.
Solution: log 25 + log 4 = log (25 × 4)
= log
100
= log
10^{2}
= 2
log 10
= 2 × 1
= 2
Example 2: Find the value of log 50. (Given log 2 = 0.3010)
Solution: log 50 = log (100/2)
= log 100 – log 2
= 2 – 0.3010
= 1.6990
Example 3: Simplify 2 log_{10} 5 + log_{10}
8 – ½ log_{10} 4
Solution:
Example 4: Express log_{10} ^{7}√72 in terms of log_{10} 2 and log_{10} 3.
Solution: