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. 2^{5} =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, a^{x} = N, then log_{a}N =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 log_{e} 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:**

Log_{e} N = 2.3026 log_{10} N or /n N = 2.3026 log_{10} N = 2.3026 log N.

Logarithm of a negative number is meaningless, because it does not exist.

Learn distance between two points.

### Fundamental theorems of Logarithm

Log_{a} mn = log_{a} m + log_{a} n

Proof: Suppose a^{x} =m and a^{y}=n Then log_{a} m =x and log_{a} n = y.

Hence a^{x} x a^{y} = mn or a^{x+y}= mn.

Or, log_{a} mn = x+y = log_{a}m + log_{a} n.

In general, we can also write that

log_{a} mnpq …. = log_{a}m+ log_{a}n+ log_{a}p+ log_{a}q+…

**Some fundamental formulas **

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