Index laws / Logarithmic laws

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Index laws



$a,b\quad \epsilon \quad { R }^{ + }$
$m,n\quad \epsilon \quad { Q }$

${ a }^{ m }\times { a }^{ n }={ a }^{ m+n }\\ { a }^{ m }\div { a }^{ n }={ a }^{ m-n }\\ ({ a }^{ { m } })^{ n }={ a }^{ mn }\\ { ab }^{ m }={ a }^{ m }\times { b }^{ m }\\ \\ { (\frac { a }{ b } ) }^{ m }=(\frac { { a }^{ m } }{ { b }^{ m } } )$

${ a }^{ 0 }=1\\ { a }^{ -n }=\frac { 1 }{ { a }^{ n } }$

If Root of a real number a in n

$a\quad \E \quad \Re$ , a>0 There are two roots if and n is even.They are equal in magnitude and opposite in sign.
 $\frac { 1 }{ { a }^{ n } }$ or  $\sqrt [ n ]{ a }$ 
$.{ a }^{ b }=\aleph$

Logarithmic laws

You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that ${ x }^{ 3 }$ × ${ x }^{ 4 }$ equals ${ x }^{ 7 }$ because you can add the exponents. There are similar rules for logarithms.

define logarithms as

${ a }^{ b }=N\quad \Leftrightarrow \quad b=\log _{ a }{ N }$
(a>0 , N >0 )


$ \log _{ a }{ MN } =\log _{ a }{ M } +\log _{ a }{ N }$

$\log _{ a }{ (\frac { M }{ N } ) } =\log _{ a }{ M } -\log _{ a }{ N } $

$ \log _{ a }{ { N }^{ p } } =p\log _{ a }{ N }$

The Change-of-Base Formula

$ \log _{ a }{ b } =\frac { 1 }{ \log _{ b }{ a } } $ 


Here $a,b,c\quad \epsilon \quad \Re $