Log question

Question

Given that e and f are integers, both greater than one, and √(e(√e(√e)=f, determine the least possible value of e+f.

                                                                                          

How to get 384 as answer?

 

This is a question in the logarithm chapter. What do I do?

Answer 1

Raising both sides of the equation to the 8th power yields

 

e⁷ =  f⁸

 

Since both e and f are integers, they can be factored as powers of the primes.   Since we are interested in a minimum type solution, there is motivation to look at the lowest prime i.e.  2  .    So let

 

e = 2ⁿ     and f = 2^m    

 

The equation then reads       2⁷ⁿ   =  2^8m      

 

This implies that   7n = 8 m.              The solution to this equation with the smallest values for n and m is

 

m = 7   and n = 8        So   e = 2⁸ =  256   and  f = 2⁷ = 128

 

Finally 256 + 128 =  384.

 

This analysis does not prove that this is the lowest possible value for e+f, but some trial and error with primes other than 2 always leads to greater values of e + f.     I am not sure why this problem was in the log section.