# Basic properties of Logarithms.

Logarithms are important throughout mathematics and science.

Establishing some basic properties or relations of Logarithms helps

us for faster computation of mathematical problems.

Some of the basic properties of Logarithms and proof of them are given below, they are also called theorems of logarithm.

1>  For any positive x ,y and a:

$log_a(xy)=log_ax+log_ay$

Proof:

Suppose: $x=a^b \quad and \quad y=a^c$

then,$log_ax=b$ and $log_ay=c$

so, $xy=a^b.a^c=a^{b+c}$

thus $log_a(xy)=log_ax+log_ay$

2> For any positive a and x:

$log_ax^p=p.log_ax$

Proof:

suppose, $x=a^m$

then $log_ax=m$

and $x^p=a^{p.m}$

thus, $log_ax^p=p.log_ax$

3> For any positive x , y and a:

$log_a(x/y)=log_ax-log_ay$

proof:

$log_a(x/y) =log_ax+log_a(1/y)$
$=log_ax+log_ay^-1$
$=log_ax+log_ay.-1$
$=log_ax-log_ay$

4> For any positive a,b and x:

$log_ax=log_bx.log_ab$

Proof:

suppose , $x=b^m$

then ,$log_bx=m$

thus ,
$log_ax = log_ab^m$
$= m.log_ab$
$= log_bx.log_ab$

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