Question

if a power b = b power c= ab, then b+c always equal to?

Answer 1

See the pic for solution

Answer 2

Given, $a^b= b^c=ab$

solving separately ,
e.g., $a^b=ab$
taking log both sides,
$loga^b=logab$
as you know, $logA^n=nlogA$
and $logMN=logM+logN$
so, bloga = loga + logb
(b - 1)loga = logb
loga^(b - 1) = logb
a^(b - 1) = b .......(1)

now, $b^c=ab$
from equation (1),
$a^{(bc - c)} = aa^{(b-1)}$
$a^{(bc-c)}=a^b$
bc - c = b [ as you know, x^m = x^n then, m =n]
bc = b + c

hence, b + c = bc