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Daca \( a,b,c\in (1,\inf) \), \( abc=8 \) si \( \log_{a}{(a-1)}+\log_{b}{(b-1)}+\log_{c}{(c-1)}=0 \), atunci aflati numerele reale \( a,b,c \).

a=b=c=2 (se foloseste AM-GM)

\( 0=\sum\log_a(a-1)\cdot1\le\sum\log_a(\frac{a-1+1}{2})^2=6-\sum\frac{1}{\log_4{a}}\le 6-\frac{9}{\sum\log_4a}=0 \)

E bine asa ?

Puteti detalia un pic, va rog? Cum folosim AM-GM? Ar putea fi, de exemplu, a<2 si atunci \( \log_{a}{a-1} < 0 \).