Question 771674
{{{1/log(a,(abc))}}}+{{{1/log(b,(abc))}}}+{{{1/log(c,(abc))=1}}}
 
Your working start is good:
{{{1/((log(abc)/log(a)))}}} + {{{1/((log(abc)/log(b)))}}} + {{{1/((log(abc)/log(c)))}}} 
={{{log((a))/log((abc))}}} + {{{log((b))/log((abc))}}} + {{{log((c))/log((abc))}}}
Then, you forget that you are adding fractions with a common denominator, and multiply the denominators instead of leaving the common denominator alone, and just adding the numerators,like this:
={{{(log((a))+log((b))+log((c)))/log((abc))}}}={{{log((abc))/log((abc))}}}={{{1}}}
That kind of confusion happens to most of us now and then.
Unfortunately, once we make a mistake like that we cannot detect it by ourselves.
We need help from a pair of fresh, unbiased eyes.