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Examples 9 to 12

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(9) Simplify :  \left [\frac{1}{\textup{log}_{x}(xyz)}+\frac{1}{\textup{log}_{y}(xyz)}+\frac{1}{\textup{log}_{z}(xyz)} \right ]

Soln.

Given expression   = \textup{log}_{xyz}\textup{ } \left (x \right ) + \textup{log}_{xyz}\textup{ } \left (y \right ) + \textup{log}_{xyz}\textup{ } \left (z \right ) \left (\therefore \textup{log}_{a}x=\frac{1}{\textup{log}_{a}x} \right )

= \textup{log}_{xyz}\textup{ }\left (x\times y\times z \right )

=   1

(10) Find the value of  \textup{log} _{10} 4 + \textup{log} _{10}  125 – \frac{1}{2} \textup{log} _{10} 25.

Soln. \textup{log} _{10} 4 + \textup{log} _{10}  125 – \frac{1}{2} \textup{log} _{10} 25.

=    \textup{log} _{10} 4 + \textup{log} _{10}  125 –  \textup{log} _{10} \sqrt{25}

=    \textup{log} _{10} 4 + \textup{log} _{10}  125 –  \textup{log} _{10} 5

=    \textup{log} _{10} \left (\frac{4\times125}{5} \right )

=      \textup{log} _{10} 100

=      2      (    \textup{log} _{10} 100 =  \textup{log} _{10} 10^{2} )

(11) Simplify : \left (2\textup{log}\frac{2}{3}+\textup{log}\frac{20}{9}-\textup{log}\frac{8}{81} \right )

Soln.   2\textup{log}\frac{2}{3}+\textup{log}\frac{20}{9}-\textup{log}\frac{8}{81}  =    \textup{log}\left (\frac{2}{3} \right )^{2} + \textup{log}\frac{2^{2}\times 5}{3^{2}} – \textup{log} \frac{2^{3}}{3^{4}}

=    \textup{log} \left (\frac{2^{2}}{3^{2}} \times \frac{2^{2}\times 5}{3^{2}} \times \frac{3^{4}}{2^{3}} \right )

\left (\because \textup{log}_{a}x-\textup{log}_{a}y=\left (\textup{log}_{a}\frac{x}{y} \right ) \right )

=     log   10

=     1                      (      log  10   =  \textup{log} _{10}10  =  1  )

(12) If  \textup{log} _{10}  2  =  0.3010  and  \textup{log} _{10}  3  =  0.4717,  find :

(i)  \textup{log} _{10} 125            (ii)  \textup{log} _{10} 13.

Soln.

(i)    \textup{log} _{10}  125

=    \textup{log} _{10} 5^{3}

=      3  \textup{log} _{10} 5

=      3  (  1  –  \textup{log} _{10} 2 )   (  \because  \textup{log} _{10} 5  =  \textup{log} _{10}  10  –  \textup{log} _{10} 2  )

=      3  (  1  –  0.3010 )

=      2.097

(ii)

\textup{log} _{10} 13.5 =    \textup{log} _{10} \frac{27}{2}              \left (\because 13.5=\frac{135}{10}=\frac{27}{2} \right )

=    \textup{log} _{10}  3^{3}  –  \textup{log} _{10} 2

=     3  \textup{log} _{10} 3  –  \textup{log} _{10} 2

=     3  ( 0.4717 )  –  ( 0.3010 )

=     1.1141

 

Best of Luck…

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