Logaritmi

Adriana Nistor · Post by **Adriana Nistor** » Thu Sep 17, 2009 1:19 pm

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 \( a,b,c \).

Marius Mainea · Post by **Marius Mainea** » Thu Sep 17, 2009 7:45 pm

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 ?

Adriana Nistor · Post by **Adriana Nistor** » Fri Sep 18, 2009 6:35 pm

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