# logarithm

What is logarithm?

The term logarithm is basically derived from two words, “logas” and “arithmos”. Logas implies ratio while arithmos means number. The logarithm of any number to a given base is the power to which the base must be raised to obtain that number.
Foe example, we know that 2 raised to power 5 is equal to 32 i.e. 25 =32. In the log form, this may be written as log 2 32 =5 i.e. logarithm of 32 to the base 2 is equal to 5. In general, ax = N, then logaN =x.

here is some basic logarithm.

### Natural logarithm

In this type of logarithms, the base is e, where

e=1+(1/1!)+ (2/2!)+ (3/3!)+…… = 2.7182818
The natural logarithm is also abbreviated as/n.ie. we may write loge N as /n N.

### Common logarithms

The common logarithm of a number is the power to which 10 must be raised in order to obtain that number i.e. in this type of logarithm, the base is 10.
The natural logarithm may be changed to common logarithm using the following relation:

Loge N = 2.3026 log10 N or /n N = 2.3026 log10 N = 2.3026 log N.
Logarithm of a negative number is meaningless, because it does not exist.

### Fundamental theorems of Logarithm

Loga mn = loga m + loga n

Proof: Suppose ax =m and ay=n Then loga m =x and loga n = y.
Hence ax x ay = mn or ax+y= mn.
Or, loga mn = x+y = logam + loga n.
In general, we can also write that
loga mnpq …. = logam+ logan+ logap+ logaq+…

Some fundamental formulas