Given log(base a)x*log(base b)x + log(base b)x*log(base c)x +log(base c)x*log(base a)x = log(base a)x*log(base b)x*log(base c)x. prove that x = abc

devide both sides by

log(base a)x*log(base b)x*log(base c)x.
to get 1/ log(base c)x + 1/log(base a)x + log(base b)x = 1

now 1/ log(base c)x = log (base x) c

so we get log (base x) c + log (base x) a + log (base x) c = 1

or aply the product rule to get

log (base x) abc = 1

so abc = x^1 = x

proved