Question

if a^x=bc,b^y=ca and b^z=ab then show that abc =a+b+c+2

Answer 1

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Dear

a^x = bc
b^y = ca
c^z = ab

x ln a = ln b + ln c
y ln b = ln c + ln a
z ln c = ln a + ln b

Let's write a for ln a , b for ln b , c for ln c

x a = b + c
y b = c + a
z c = a + b

(x+y+z)abc = (b+c)bc + (c+a)ca + (a+b)ab

xyz abc = (b+c)(c+a)(a+b)

(xyz - (x+y+z))abc =

[ bca+bcb+baa+bab+cca+ccb+caa+cab ]
-
[ bbc+cbc+cca+aca+aab+bab]

= 2abc

xyz - (x+y+z) = 2

Answer 2

hopefully this will be able to help you