+0  
 
0
143
3
avatar+11 

Let a and b be positive real numbers such that a^b = b^a and b = 9a. Then a can be expressed in the form[ m sqrt (n)] where m and n are positive integers, and n is as small as possible. Find m + n

 

 

 

Thanks Loves

 Apr 4, 2020
 #1
avatar+21958 
+1

              ab  =  ba 

      log(ab)  =  log(ba)

     b·log(a)  =  a·log(b)

 

Since  b = 9a:

       9a·log(a)  =  a·log(9a)

Divide both sides by a:

         9·log(a)  =  log(9a)

           log(a9)  =  log(9a)

                  a9  =  9a

 

Either a = 1  (which won't work)

 

or:         a8  =  9

               a  =  91/8

               a  =  (32)1/8

                a  =  31/4

 

and  b  =  9·31/4

 

Which means that I can't get the answer that you want!

 Apr 4, 2020
 #2
avatar+112415 
+2

a^b  =  b^a

 

Take  the log of each side

 

b log a =  a log b

 

(b/a)  = log b  / log a

 

[( 9a)  /a ] =  log b  /log a

 

9  = log b / log a

 

Which implies  that

 

log a b  = 9

 

So

 

a^9  = b

 

a^9  =  9a

 

a^8  =  9      take  the  8th root of  each side

 

a =  9^(1/8) = (3^2)^(1/8)  =  3^(2/8)  =  3^(1/4)  =  4√ 3 

 

This is what geno  found and I'm assuming that it is the  form you  want

 

So

 

m + n  =   7

 

 

cool cool cool

 Apr 4, 2020
 #3
avatar+11 
0

Thanks You guys Sooooo much

 

 

WITH MUCH LOVE

- C

 Apr 5, 2020

42 Online Users

avatar
avatar