9th Class Mathematics Related to Competitive Exam Logarithms

LOGARITHMS

FUNDAMENTALS

• Logarithm:- Let a be a positive real number other than 1 and \[{{a}^{x}}=m(where\,m>0)\]then x is called the logarithm, of m to the base a and written, as \[lo{{g}_{a}}\]m.

Example:- \[\left( I \right){{10}^{4}}=10000\]

                        \[\Rightarrow lo{{g}_{10}}=10000=4\]

(II) \[If{{3}^{-3}}=\frac{1}{27}\Rightarrow {{\log }_{3}}\frac{1}{27}=-3\]

LAWS OF LOGARITHMS 

• \[\text{(I)}\,\,{{\log }_{a}}(mn)={{\log }_{4}}m+{{\log }_{a}}n\]
• \[\text{(II)}\,\,{{\log }_{a}}\frac{m}{n}={{\log }_{4}}m-{{\log }_{a}}n\]
• \[\text{(III)}\,\,{{\log }_{a}}a=1\]
• \[\text{(IV)}\,\,{{\log }_{a}}1=0\]
• \[\text{(V)}\,\,{{\log }_{a}}({{m}^{p}})=P({{\log }_{a}}m)\]
• \[\text{(VI)}\,\,{{\log }_{a}}m=\frac{1}{{{\log }_{m}}a}\]
• \[\text{(VII)}\,\,{{\log }_{a}}m=\frac{{{\log }_{b}}m}{{{\log }_{b}}a}=\frac{\log m}{\log a}\]
• \[\text{(VIII)}\,\,{{\log }_{{{a}^{k}}}}b=\frac{1}{k}{{\log }_{a}}b\]