Logarithm | log rules, formula and examples

Logarithm Definition Law of Logarithm Product Rule Quotient Rule Power Rule Some other rules of logarithm Special log properties

Logarithm

Introduction

 Logarithm was discovered in 17th century in Europe as to simplify an mathematical calculations . It was a Greek word which means arithmetic, ratio, and proportion. If someone asked us, "multiply 10'times 2" what would do you simply multiple the 2 Ten times like that, 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 = 1,024.  If I said that write it in another way so what you would do you us exponential expression by these you can write it as

2¹⁰ = 1024

If I said that there is another way to solve or write this equation, what is the method? It was Logarithm an another way of thinking to solve exponential equations.

What is Logarithm ?

In previous classes we study exponential, we known that if 6'raised to the power 3 the resultant is 216. This is known as exponential expression. written as,

6³ = 216

Let assume that someone asked us that if 6'raised to which power to get 216? . The answer was 3. This is expressed by the logarithmic equation.written as,

log₆ 216 = 3

we can read this equation as log of 216 at base 6 is 3, or log base 6 of 216 is 3.
Relation ship between exponential and logarithm expression.

6³ = 216  ⇔  log_a 216 = 3

• 6 is the Base
• 3 is the power(index - exponent)
• 216 is the resultant or argument

Use of logarithm

Logarithm is a revers exponent form because of this reason, they are very useful and helpful for solving and understanding exponential equations. Sometimes we come across Logarithm expression is very interesting to learn and understand, because it expresses numerical expressions involving multiplication, division, and rational power of large numbers. Logarithm are very helpful in making lengthy and difficult calculations easy. It is very common around us in the world. Here we introduce the concept of logarithms, and apply the technique to simplify numerical expressions.

Definition

Assume that a is a positive real number, other then 1(a≠1) and x is a rational number, then the exponential expression is a^x = n , So that we says that logarithm of n at base a is x or x is the log of n to the base a, written as log_an = x.
Thus,

a^x = n ⇔ log_a n = x

log_an = x is a exponential expression, and
a^x = n is a logarithmic expression

Examples from above definition:

1. 
2. 
3. 
4. 
5. 
6. 

Remember that : Any positive real number a, we know that and .

∴  and  .

Decimal Logarithm :

All of our examples we have used whole numbers like 2, 3,or 5 etc. but log have decimal values which we discussed in our previous examples (5.) where, we have seen that log 2 at base 10 is 0.3010. 

for example :

 

Some other examples of the above definition that:

1.)  Find the real number x ?
from the above definition that :
 A real number x ⇔
Thus,

x=5
So that,
            A real number is 5 
2.)   Find the real number x ?
from the above definition that :
 A real number x ⇔ 
Thus,

x = 1/2

So that,
             A real number is 1/2

3.)    Find the real number x ?
from the above definition that :
 A real number x ⇔ 
Thus,

x=-3
So that,
            A real number is -3

4.)   Find the real number x ?
from the above definition that :
 A real number x ⇔ 
Thus,

x=1
So that,
           A real number is 1
Remember that : "log" is the abbreviation of the word "logarithm"

Laws of Logarithm

The basic rules of logarithm can expressed in many different ways are collectively named as "fundamental Laws of logarithm" .
There are Four fundamental fundamental of logarithm .

1st law :   Product Rule,

2nd law : Quotient Rule, and

3rd law :  Power Rule.

Product Rule :

The logarithm of the product two numbers or more then two numbers is equal to the sum of their logarithms .

Proof :

If m,n are positive real numbers, then the product of  them is (mn)

Let  

and  . Then, 

[by  definition of logarithm]

                                                            [by law of exponent]

                                            [by definition of log]

               []

Quotient Rule :

The logarithm of the ratio of two numbers or more then two numbers is equal to the difference of their logarithms.

Proof :

If  m and n are positive rational numbers,  then,

       [by definition of log]

 

                                                   [by exponential law]

                               [by the definition of log]

Note : if x, y are real numbers, (where xy > 0) and a is a positive real numbers (where a ≠ 1), then

Power Rule :

If m,n are positive rational number, and a is a positive real number (where a > 0 and a ≠ 1), 

Proof :

            [by definition of logarithm]

Some other Rules of Logarithm :

There are six other rules of logarithm .

First Law :

If a,b are positive real number (where a ≠ 1, b ≠ 1), and m is positive rational number, then

Proof :

           [Taking log to the base b]

Note :

On replacing b by m in the above equation result, we get

Second law :

The log of 1 to any base is always Zero.

Proof :

We have, (where a > 0)

[Therefore, by definition of log]

Third Law :

The log of any positive number to the same base is always one.

Proof : 

For any positive number a,  

We know that,

 Therefore, by definition of log,

Fourth Law :

If 'n' is a positive rational number, and 'a' is  a positive real number, other then 1, Then

Proof :

Fifth Law :

If 'n' is a positive rational number, 'a' is a positive real number, (Where a ≠ 1), Then

Proof :

                               [Substituting the value of x and y]

Replacing a = n in the above result, we obtain

Six Law :

If x'a' is a positive real number, (Where a ≠ 1) and x,y are positive rational number, Then 

Proof :

We know that :

Special log properties :

Various types of logarithms base, but here we will discuss about two only which are widely used in logarithm. Most of the calculation is done in only this two.

Common Logarithm :

A log whose base is 10 (x log at base 10 ) is called common logarithm . When we write the equation, we ignore the base of logarithm.

for example :

Natural logarithm :

 A log whose base is e (x log at base e ) is called natural log. The value of e = 2.718282, where e is called Euler's number. In engineering it is written In(x).

There was always a confusion around us about Common log and Natural log. Because in mathematics "log" is used while in engineering "In" is used. this can generates confusion around students.
For example :

log (86)

In engineering students thinks that it is a (log 86 at base 10) and mathematician thinks (log 86 at base e ) this type of confusion around students.