show that b^c×c^b=c^a×a^c=a^b×b^a

Question

if a(b+c-a)/loga=b(c+a-b)/loga=c(a+b-c)/logc show that b^c×c^b=c^a×a^c

Answer 1

Do you mean "if a(b+c-a)/loga=b(c+a-b)/logb=c(a+b-c)/logc show that b^c×c^b=c^a×a^c"?

Let r=a(b+c-a)/log(a) and therefore the other two expressions.

log(a^r)=a(b+c-a), log(b^r)=b(c+a-b), log(c^r)=c(a+b-c).

log(a)=a(b+c-a)/r, log(b)=b(c+a-b)/r, log(c)=c(a+b-c)/r.

log(b)-log(a)=(a-b)(a+b-c)/r=(a-b)log(c)/c. Therefore c(log(b)-log(a))=(a-b)log(c) (1)

If b^cc^b=c^aa^c, clog(b)+blog(c)=alog(c)+clog(a), c(log(b)-log(a))=(a-b)log(c) (2)

(1) and (2) are identical, so b^cc^b=c^aa^c⇒c(log(b)-log(a))=(a-b)log(c).