Using the power property of logarithms to compare the numbers 600^601 and 601^600. Which is larger?

Question

similarly of the two numbers 1/99^100 and 1/100^99 which is smaller

Answer 1

 

600^601 ?  601^600

 

Applying log to both sides of the "?" symbol, we have:  Y = X^A       log Y  =  A * log X

 

601*log(600)  ?  600*log(601)    

 

Comments: The log is a curve that have a very subtle change for larger numbers (asymptotic behaviour). Then, it is expected that log(600) and log(601) have practiaclly the same value.

Then, you really has to compare the numbers 601 and 600....and it is obvious that the number 601 is larger than 600.

 

The solution to your problem is: 600^601 is larger than 600^601

 

For the second case:    99^100  ?   100^99

 

                                100 log(99)  ?   99 log(100)       100*(1.995)  ?   99*(2)       199.56 > 198

 

                                Then, 99^100  is larger than 100^99

 

But you are comparing:   1/(99^100)  and 1/(100^99)   In this case due to both numbers are denominators of fractions, it is clear that the larger fraction is one where the denominator have the smallest  value. Based on your comparison above,   

 

1/100^99   is larger than  1/99^100     because    100^99 is smaller than 99^100

 

I hope it helps.