11. (Ufjf-pism 1 2017) Sejam a, b, c e d números reais positivos, tais que blog a = 5, blog c = 2 e blog d = 3. O valor da expressão (2/5)c^(3/a) ...

11. (Ufjf-pism 1 2017) Sejam a, b, c e d números reais positivos, tais que blog a = 5, blog c = 2 e blog d = 3. O valor da expressão (2/5)c^(3/a) * d^(2/b) é igual a:

blog a = 5
blog c = 2
blog d = 3
(2/5)c^(3/a) * d^(2/b) = (2/5)*c^(3/a)*c^(3logd/5logc) = (2/5)*c^(3/a + 3logd/5logc) = (2/5)*c^(3/a + 3log2/5) = (2/5)*c^(3/a)*2^(3/5)
(2/5)*c^(3/a)*2^(3/5) = (2/5)*c^(3/a)*2^(3bloga/5log2) = (2/5)*c^(3/a)*2^(3*5/5) = (2/5)*c^(3/a)*8
(2/5)*c^(3/a)*8 = (2/5)*2^(3logc/5log2)*8^(5/3) = (2/5)*2^(3*2/5)*8^(5/3) = (2/5)*2^(6/5)*2^(5/3) = (2/5)*2^(23/15)*c^(3/a)
(2/5)*2^(23/15)*c^(3/a) = (2/5)*2^(23/15)*c^(3log2/5loga) = (2/5)*2^(23/15)*2^(3/5) = (2/5)*2^(26/15)
(2/5)*2^(26/15) = 2^(1/15)
O valor da expressão é igual a 2^(1/15).
a) 1
b) 2
c) 3
d) 4
e) 0