Logarithm
Logarithms were developed to solve the
complicated problems in very simple and easy way. In Science, we need to
calculate the 8th or 10th (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 5th 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 πr3
= 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 ax = 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 loga M = x.
For example, 23 = 8, then power ‘3’ is called the logarithm of the number 8 to the base 2, and written as log2 8 = 3.
It is to be noted that ax = M
is called the exponential form and loga M = x is
called the logarithmic form.
The following examples will explain clearly about Exponential form and Logarithm form.
|
Exponential Form |
Logarithm Form |
|
26 = 64 |
log2 64 = 6 |
|
33 = 27 |
log3 27 = 3 |
|
44 = 256 |
log4 256 = 4 |
|
100 = 1 |
log10 1 = 0 |
|
101 = 10 |
log10 10 = 1 |
|
102 = 100 |
log10 100 = 2 |
|
10-1 = 0.1 |
log10 0.1 = -1 |
|
10-2 = 0.01 |
log10 0.01 = -2 |
Logarithm Rules or Laws of Logarithm
1. Product Rule:
loga (MN) = loga M + loga
N
2. Quotient Rule:
loga (M/N) = loga M - loga
N
3. Power Rule:
loga (MN) = N.loga
M
4. Zero Rule:
The logarithm of 1 to any base is zero, i.e., logx
1 = 0.
5. Identity Rule:
The
logarithm of a number to the same base is equal 1,
i.e., logx
x = 1.
6. Inverse Property of Logarithm:
loga (ak) = k.
7. Inverse Property of Exponent:
aloga(k)
= k.
8. Change of Base Formula:
loga x = logc
x/logc 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, loga MN = loga
M + loga N
Let loga M = x and loga
N = y.
Changing each of these logarithms into
exponential form, we get
ax = M ……… (i)
ay = N ……… (ii)
Multiplying equations (i) and (ii), we
get
MN = ax. ay
MN = ax + y (By laws of exponents)
Again, changing it back to logarithmic
form, we get
loga MN = x +
y
Putting the values of x and y, we get
loga MN = loga M + loga 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,
loga M/N = loga
M - loga N
Let loga M = x and loga
N = y.
Changing each of these logarithms into
exponential form, we get
ax = M ……… (i)
ay = N ……… (ii)
Dividing equation (i) by (ii), we get
M/N = ax/ ay
M/N = ax - y (By laws of exponents)
Again, changing it back to logarithmic
form, we get
loga M/N = x -
y
Putting the values of x and y, we get
loga M/N = loga M - loga 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, loga
(MN) = N.loga M
Let loga M = x
Changing this into exponential form,
we get
ax = M
Raising both sides to the power N, we
get
(ax)N = (M)N
aNx
= (M)N (Using
power of power law of exponents)
Changing this back into logarithmic
form, we get
loga (M)N = Nx
Putting x = loga M back
here, we get
loga (M)N = N.loga
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
102
= 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 log10 5 + log10
8 – ½ log10 4
Solution:
Example 4: Express log10 7√72 in terms of log10 2 and log10 3.
Solution:
