Máme xoy = x ^ (xlog_e y), forall x, yin [1, oo). Nájdite x pre x o x o x = 125?

odpoveď:

#x = e ^ root (4) (3 log 5) #

Vzhľadom k tomu, že pre #x> 0 rArr x = e ^ (log x) #

a definovanie # x @ y = e ^ (logx logy) #

máme

# x @ x @ x = e ^ (Log (e ^ (Log (2x ^) Logx)) Logx) = ((e ^ (Log ^ 2x)) ^ Logx) ^ Logx #

potom

# ((E ^ (Log ^ 2x)) ^ Logx) ^ Logx = 5 ^ 3 #

teraz #log # na obe strany

#logx log (e ^ (Log ^ 2x)) ^ Logx = log ^ 2x log (e ^ (Log ^ 2x)) = log ^ 4x = 3 log 5 #

potom

#log x = root (4) (3 log 5) # a

#x = e ^ root (4) (3 log 5) #