Logarithm

Logarithm is closely connected to exponent. It is just inverse of exponent. If a^x= N then the Logarithm is:

x=log_aN

'x' is called logarithm of 'N' on base 'a'. This is stated as "log to base a of N equals x".

The base(here a) of the logarithm will always be a positive number and not equal to 1.

For example,

2³= 8     log₂8 = 3

3⁴= 81     log₃81 = 4

Logarithms can also be negative:

1/2¹= 2^-1     log₂1/2 = -1

1/5²= 5^-2     log₅1/25 = -2

But look at the example, log₄^(–16) = ?. What will be the logarithm?

If log₄^(–16) =x, then 4^x = -16.  What power of 'x' could possibly turn a positive 4 into a negative 16? This is impossible. So, in the exponential equation, a^x=N,  'N' will be any real positive number. We are raising a positive number to an exponent and so there is no way that the result can possibly be anything other than another positive number.

For any base a, log_aa = 1 and log_a1 = 0, since a¹ = a and a⁰ = 1.

Laws of Logarithm:

log_a(MxN) = log_aM + log_aN

log_a(M/N) = log_aM - log_aN

log_aM^P  = P log_aM

log_aP√M  = 1/P log_aM

There are 2 types of logarithms. The Common Logarithm and Natural Logarithm.

Logarithm which has base 10 is considered as Common Logarithm.  Henry Briggs first compiled 'Tables of Logarithms' in 1617. If base is not mentioned then it should be understood that base is 10. So log 4 means log₁₀4.

Before H. Briggs, John Napier, first introduced Logarithms in 1614 as a means to simplify calculations.The natural logarithm has the number 'e'  as its base. Here e ≈ 2.71828.

'e' is an irrational number. It is often called Euler's number. Napier used 'e' as base in Natural Logarithm. The natural logarithm of 'x' is generally written as ln x or log_ex.

Logarithm's main application was to reduce calculation or simplify calculation. Logarithm is widely used by scientists and engineers, and others to perform computations more easily. In Computer Science, binary logarithm(base 2 ) is used.