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Examples 5 to 8

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(5) Evaluate :  \textup{log}_{2} ( \textup{log}_{5}  625 ).

Soln.  \textup{log}_{2} ( \textup{log}_{5}  625 ) =    \textup{log}_{2} ( \textup{log}_{5}  5^{4} )

=    \textup{log}_{2} ( 4  \textup{log}_{5} 5 )

=    \textup{log}_{2} 4

=  \textup{log}_{2}   2

=    2  \textup{log}_{2} 2     =    2

(6) If  log10 2  =  0.30103, then find the number of  digits in 2^{100}.

Soln. \textup{log} _{10} (2^{100})  =  100 \times  \textup{log} _{10} 2  =  100 \times 0.30103  =  30.10.

\therefore Its characteristic is  30.

Hence, the number of  digit in 2^{100}    is 31.

(7) Evaluate :  \textup{log} _{z} x . \textup{log} _{w} y . \textup{log} _{x} z . \textup{log} _{y}  w.  

Soln.

\textup{log} _{z} x . \textup{log} _{w} y . \textup{log} _{x} z . \textup{log} _{y}  w

= \frac{\textup{log }x}{\textup{log }z} \times \frac{\textup{log }y}{\textup{log }w} \times \frac{\textup{log }z}{\textup{log }x} \times \frac{\textup{log }w}{\textup{log }y}

=        1

(8) If   log  [ 99 + \sqrt{x^{2}-12x+36} ]  =  2, find x.

Soln. \textup{log} _{10}  [ 99 + \sqrt{x^{2}-12x+36}  ]  =  2  =  \textup{log} _{10} 100

\therefore 99 + \sqrt{x^{2}-12x+36}  = 10^{2}

\therefore 99 + \sqrt{x^{2}-12x+36}  =  100

\therefore \sqrt{x^{2}-12x+36}=  1

\therefore x2 – 12x + 36  =  1

\therefore x2 – 12x + 35  =  0

\therefore ( x – 7 ) ( x – 5 )  =  0

\therefore x  =  7   or   x  =  5.

 

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