Logarithm

Logarithm

History:

The scottish mathematian John Napier discovered logarithm. His discovery in logarithm was published in 1614. But the logarithm he  had discovered was very different from the modern logarithm.

  After John Napier , another mathematian Henry Briggs introduced base-10 logarithm. Which was very  easier to use. After that, many other mathematian contributed their theories about logarithm. In 1730, Euler defines exponential function (eˣ) and natural logarithm. The developments of modern logarithm was extended in 18-th century. Till now mathematians are updating logarithm and  introducing new theories.

Introduction:

We know that, if "a" and "x" be real , a not equal to 0, then a  and x are called the base and power/exponent/index of a in aˣ.

  Now, we can get the value of M in aˣ=M, if the values of a and x are given. As for example, if a=2, x=3 then, M= 2³=8.

 Again we can find the value of a from  aˣ=M, if x and M are given. As for example, of x=2, M=4, a=+2 or -2.

   But if the values of a and M are given , we can't get the value of x  easily from aˣ=M.

  As for example, if a=3, M=9, we get, x=2 very easily but if a=2, M=5, we are unable to get the value of x , easily by algebraic methods. In this we will use a different method which is called Logarithm.

Definition of Logarithm:

If aˣ=M, (a>0,M>0, a not equal to 1) then x is called the Logarithm of M to the base a ,and expressed as: x=logₐ M.

Converse is also true.

    Note:

(1) If we do not specify/mention the base, Logarithm is meaningless.

(2) The values of a logarithm of a number with respect to different bases will be different.

(3) The value of Logarithm for a negative number is undefined. i.e, if aˣ=-M(a and M both are positive, then value of x will be imaginary.

(4) The logarithm of 1 with respect to any base a(not equal to 0) is always 0.

(5) If a and M both are same positive number, the the value of x or value of Logarithm will be 1(as, log a a=1).

 (6) If, x= logₐ M , then , aᴸᵒᵍₐᴹ=M.

(7) Logarithm of zero is undefined. 

Laws of Logarithm:

(1) log ₐ (MN) = log ₐ (M) + log ₐ (N).

(2) log ₐ (M/N)= log ₐ (M) - log ₐ (N).

(3) log ₐ (M) = log ₓ (M) × log ₐ (x).

(4) log ₐ (M^n)= n×log ₐ (M).

(5) log ₓ (a) = 1/{log a (x) }.

(6) log ₐ (x) × log ₓ (a)= 1.

Where, M, N, a,x>0, a and b not equals to 1, n be any real number.

Some problems:

   problem:1

    If log ₓ (243)=10, then find the value of x.

     
        we have, 243= 3⁵.
         now, log ₓ (243) = 10
          or, x¹⁰ =243 = 3⁵
          or, x² = 3
           or, x =√3.
   so, the value of x is √3.

   problem:2
    If log ₇¹/² (343) = x , then what is the value of x?

       we have, 343=7³.
        now, log ₇¹/² (343) = x
            or, 7ˣ/² = 343=7³
             or, x/2 =3
              or, x=6.
  so, the value of x is 6.

  problem:3
   Calculate, log ₂ log ₂ log ₂ (16) =?
     we have, log ₂ log ₂ log ₂ (2⁴)
                    = log ₂ log ₂ (4 log ₂ 2)
                    = log ₂ log ₂( 2²)
                    = log ₂ (2 log ₂ 2)
                    = log ₂ 2 = 1.       [log ₐ a =1]

If you find out any incorrect information or know anything more about this , please write it in the comment section!