Logarithm is a fundamental mathematical concept of algebra and it was invented to perform some operations like multiplication or division of large numbers and very smaller numbers in digits easily. Similarly, it is also helpful to us in simplifying the completed functions. Logarithm is abbreviated as log. Mathematically the logarithmic operation is represented with either log or ln symbols as per the application.
Now we have advanced technology to determine multiplication and division of any numbers without putting strain on brain by using scientific calculators and computer programs. In earlier days, there was no calculator but scientists were used this math concept to solve their problems. You may get one doubt i.e why we need to learn logarithms concept through we have calculators now. Of course we have but our calculators cannot simplify the math functions’ operation. In other words, calculators can only deal numbers not functions.
Definition of Logarithm
If a and n are positive real numbers but the value of a should not equal to 1, x is a rational number such that ax = n then we consider logarithm of n to base a is x. In other words, x is the logarithm of n to the base a.
Explanation of Definition of Logarithm
Understanding this definition of logarithm is little bit difficult but you can understand this definition with mathematical explanation. We know the value of a belongs to positive real numbers (R+) group and can express it in below form where the symbol ∈ represents belongs to.
a ∈ R+
Fine but we should exclude the value 1 from the positive real numbers in order to satisfy our assumption of a. So, we further write the above form into below form.
a ∈ R+ - {1}
Similarly, just like a, the value n also belongs to positive real number but this n also considers the value 1.
n ∈ R+
The both a and n satisfy the principle of logarithm ax = n
Now the value of x is logarithm of n with respect to base a. mathematically; we write it as stated below.
loga n = x
If we substitute the value of x i.e loga n in ax then ax should be satisfied the value n. In other words, we know that ax = n. Substitute x = loga n in ax.
ax = aloga n = n
This assumption is known as fundamental logarithmic identity.
Classification of Logarithms
Logarithms can be classified into two types based on the base of the logarithm.
If base value a is 10 that is called common logarithm and if the value of base (a) is e (2.718…..) then it is defined Natural Logarithm.