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Properties of Logarithms

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☼ Properties of  Logarithms  :

  • \textup{log}_{a} \textup{}\left (xy \right ) = \textup{log}_{a}\textup{ }x + \textup{log}_{a}\textup{ }y
  • \textup{log}_{a}\textup{ }\left (\frac{x}{y} \right ) = \textup{log}_{a}\textup{ }x-\textup{log}_{a}\textup{ }y
  • \textup{log}_{b}\textup{ }a^{m}  =  m\textup{ }\textup{log}_{b}\textup{ }a
  • \textup{log}_{a}\textup{ }1=0
  • \textup{log}_{a}\textup{ }a=1\textup{ } \left (\because a^{1}=a \right )
  • \textup{log}_{a}\textup{ }x=\frac{1}{\textup{log}_{x}\textup{ }a}
  • a^{\textup{log }x }  =  x^{\textup{log }a}
  • b^{\textup{log}_{ b}\textup{ }a} = a
  • \textup{log}_{b}\textup{ }a=\frac{\textup{log}_{c}\textup{ }a}{\textup{log}_{c}\textup{ }b} . So, \textup{log}_{b}\textup{ }a=\frac{\textup{log a}}{\textup{log b}}
  • Sign  of   logarithm : –

\large \mapsto If  0 < x < 1, then  log x  <  0.

\large \mapstoIf  x  =  1, then  log x  =  0.

\large \mapsto If  x  >  1, then  log x  >  0.

That is .

\textup{log}_{y}x=\frac{\textup{log x}}{\textup{log y}} 0 < x < 1 x  =  1 x  >  1
0 < y < 1 log x  = \frac{-ve}{-ve}

=  + ve

 log x  = \frac{0}{-ve}

=    0

 log x  = \frac{+ve}{-ve}

=  – ve
y  =  1 log x  = \frac{-ve}{0}

=  –\infty

 log x  =  \frac{0}{0}

= \infty

 log x  =\frac{+ve}{0}

= \infty
y  \geq 1 log x  =\frac{-ve}{+ve}

=  – ve

 log x  =\frac{0}{+ve}

=   0

 log x  =\frac{-ve}{+ve}

=  + ve

\textup{log}_{10} \textup{ x} contains of  two parts, the integral part is known as Characteristic and the decimal part is known as  Mantissa. \overline{1}

Number

(  x )

 log x Characteristic Mantissa
2135 \textup{log} _{10} 2135  =  3.3285 3 0.3285
213.5 \textup{log} _{10} 213.5  =  2.3285 2 0.3285
21.35 \textup{log} _{10} 21.35  =  1.3285 1 0.3285
2.135 \textup{log} _{10} 2.135  =  0.3285 0 0.3285
0.2135 \textup{log} _{10} 0.2135  =.3285 – 1 0.3285
0.02135 \textup{log} _{10} 0.02135  =  .3285 – 2 0.3285
0.002135 \textup{log} _{10} 0.002135  = .3285 – 3 0.3285

 

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