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Examples(1 to 4)

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(1) Evaluate :  (i)    \textup{log} _{5} 125                (ii)    \textup{log} _{2} \textup{ }\frac{1}{32}                         (iii)  \textup{log} _{1000}\left ( 0.1 \right )

Soln. (i) \textup{log} _{5} 125   =  \textup{log} _{5}  5^{3}   =   3 \textup{log} _{5}  5    =  3  ·  (1)  =  3

(ii) \textup{log} _{2} \textup{ }\frac{1}{32}      =  \textup{log} _{2} 2^{-5}   =    – 5 ·  \textup{log} _{2} 2    =   – 5 (1)   =  – 5

(iii) \textup{log} _{1000}\left ( 0.1 \right )   =  \frac{\textup{log 0.1}}{\textup{log 1000}} =  \frac{\textup{log }10^{-1}}{\textup{log }10^{3}}  =  \frac{-1(\textup{log 10})}{3(\textup{log 10})}  =   –\frac{1}{3}

(2) If    \textup{log} _{16}  x   =  2\frac{1}{4} ,   find  x.

Soln. \textup{log} _{16}  x   =  2\frac{1}{4} =  \frac{9}{4}  \large \Rightarrow   x  = \left (16 \right ) ^{\frac{9}{4}}   =  \left (2^{4} \right )^{\frac{9}{4}}  =  2^{9} =  512.

(3) Evaluate : 49^{\textup{log}_{7}\textup{ }5}

Soln. 49^{\textup{log}_{7}\textup{ }5} =  7^{\textup{2 log}_{7}\textup{ }5} = 7^{\textup{log}_{7}\textup{ }25} = 25        \left (\because b^{\textup{log}_{b}\textup{ }}=a \right )

(4) Evaluate : \textup{log}_{16}\textup{ 64}-\textup{log}_{64}\textup{ 16}    and \textup{log}_{2}\textup{ 5}\times \textup{log}_{25}\textup{ 8}

Soln.  (i) \textup{log}_{16}\textup{ 64}-\textup{log}_{64}\textup{ 16}

=\frac{\textup{log 64}}{\textup{log 16}}  –  \frac{\textup{log 16}}{\textup{log 64}}

=\frac{\textup{6 log 2}}{\textup{4 log 2}}  – \frac{\textup{4 log 2}}{\textup{6 log 2}}

=\frac{3}{2}-\frac{2}{3} = \frac{5}{6}

(ii) \textup{log}_{2}\textup{ 5}\times \textup{log}_{25}\textup{ 8}

=\frac{\textup{log 5}}{\textup{log 2}}  \times  \frac{\textup{log 8}}{\textup{ log 25}}

=\frac{\textup{ log 5}}{\textup{ log 2}}  \times \frac{\textup{3 log 2}}{\textup{2 log 5}}

=\frac{3}{2}

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