+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
+23245 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!
+128570 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
