sir,are log2 and log100 irrational or rational?justify your answer.

Asked by raoss1967 | 1st Dec, 2015, 09:53: PM

Expert Answer:

C o n s i d e r space 2 cross times 2 cross times 2 cross times 2 cross times 2 equals 32 R e w r i t i n g space t h e space a b o v e space n u m e r i c space e x p r e s s i o n comma space w e space h a v e comma 2 to the power of 5 equals 32 L o g a r i t h m space o f space 32 space t o space t h e space b a s e space 2 space i s space 5. rightwards double arrow l o g subscript 2 32 equals 5 T h i s space i s space t h e space c o n c e p t space o f space log a r i t h m s. N o w space c o n s i d e r space y o u r space e x a m p l e. Y o u space h a v e space n o t space m e n t i o n e d space t h e space b a s e space o f space log a r i t h m. C o n s i d e r space log subscript 2 2. W e space k n o w space t h a t space 2 to the power of 1 equals 2 comma space a n d space h e n c e comma space l o g subscript 2 2 equals 1 W e space k n o w space t h a t space 1 space i s space a space r a t i o n a l space n u m b e r. B u t space i f space y o u space t a k e space log a r i t h m space o f space 2 space w i t h space b a s e space 10 comma space log subscript 10 2 equals 0.3010..... H e r e comma space space l o g subscript 10 2 space i s space i r r a t i o n a l.  C o n s i d e r space t h e space n u m b e r space log 100. W e space k n o w space t h a t space 10 squared equals 100 T h u s space log subscript 10 100 equals 2 comma space w h i c h space i s space r a t i o n a l. B u t space i f space y o u space t a k e space log a r i t h m space o f space 100 space t o space t h e space b a s e space 2 comma space w e space h a v e comma log subscript 2 100 equals fraction numerator log subscript 10 100 over denominator log subscript 10 2 end fraction equals fraction numerator 2 over denominator 0.3010.... end fraction equals 6.6438561.... comma space w h i c h space i s space i r r a t i o n a l.  S o space log a r i t h m space o f space n u m b e r space m a y space b e space r a t i o n a l space o r space i r r a t i o n a l space d e p e n d i n g space u p o n space t h e space b a s e space t h a t space w e space a r e space c o n s i d e r i n g. H o p e space t h i s space i s space c l e a r space t o space y o u...............

Answered by Vimala Ramamurthy | 2nd Dec, 2015, 10:51: AM

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