(Above evaluations are done using common logarithms, that is logarithm to base 10. Since the natural logarithms and the common logarithms differ by only a constant, the conclusions arrived at do not change.)
Proof 2:
Using the index law a^mn = (a^m)^n , we write
100² = (10²)² = 10^4 ………………..…………………..(A)
2^100 = 2^(25x4) = (2^25)^4…………………………..(B)
On comparing (A) and (B), we find the exponents in both the equations are the same 4. Now look at their bases. Base in (A) is 10 and that in (B) is 2^25. Since 2^25 can be written as 2^4 x 2^21 = 16 . 2^21 , it follows that
Answers & Comments
Answer:
2^100 is greater
Proof 1:
log (2^100) = 100 log2 = 100 x 0.3010 = 30.10…………………………………………(1)
log 100² = 2 log 100 = 2 log 10² = 2.2 log 10 = 4…………………………………….(2)
It is obvious from Equations (1) and (2) that
2^100 is greater than 100² (Proved)
(Above evaluations are done using common logarithms, that is logarithm to base 10. Since the natural logarithms and the common logarithms differ by only a constant, the conclusions arrived at do not change.)
Proof 2:
Using the index law a^mn = (a^m)^n , we write
100² = (10²)² = 10^4 ………………..…………………..(A)
2^100 = 2^(25x4) = (2^25)^4…………………………..(B)
On comparing (A) and (B), we find the exponents in both the equations are the same 4. Now look at their bases. Base in (A) is 10 and that in (B) is 2^25. Since 2^25 can be written as 2^4 x 2^21 = 16 . 2^21 , it follows that
2^100 is greater than 100² (Proved)