Logarithm: Definition, Logarithm Rules, Derivation

# Logarithm: Definition, Logarithm Rules, Derivation

## 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 + logN

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 - logN

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.

## 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:

Related Topics:

What is an Addend in Maths

Minuend and subtrahend

Multiplicand and multiplier

Dividend, divisor, quotient and remainder

Natural numbers

Whole numbers

Properties of rational numbers

Are all integers rational numbers?

Find five rational numbers between 3/5 and 4/5