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Question

if log a : log b : log c=(b-c):(c-a):(a-b) then which one is correct


a^bb^cc^a=1 , a^ab^bc^c=1 , a^1/ab^1/bc^1/c=1

Answer 1

log a : log b : log c=(b-c):(c-a):(a-b)

I interpret this to mean that

log a / log b = (b-c)/(c-a)  and

log b / log c = (c-a)/(a-b) and so

log a / log c = (b-c) / (a-b)

The ratio assure us that none of the differences are 0.

The first equation is that log_b a=(b-c)/(c-a) or that b^(b-c)/(c-a) = a  or b^b-c = a^c-a

The second equation, inverted says that

log_b c = (a-b)/(c-a) or that b^(a-b)/(c-a) = c  or that b^a-b = c^c-a

Multi[plying these last two together b^a-b × b^b-c =(ac)^c-a or that

b^a-c = (1/ac)^a-c which seems to me to make abc=1.  That's as far as I got.