(log(x akar(x)) - log(akar(y)) + log(x/y^2))/(log(x/y)) =

5/2

Pembahasan

Mari kita sederhanakan ekspresi logaritma tersebut:

Ekspresi awal: (log(x \sqrt{x}) - log(\sqrt{y}) + log(x/y^2)) / log(x/y)

Gunakan sifat-sifat logaritma:
1. log(a*b) = log(a) + log(b)
2. log(a/b) = log(a) - log(b)
3. log(a^n) = n*log(a)
4. log(\sqrt{a}) = log(a^{1/2}) = (1/2)*log(a)

Sederhanakan bagian pembilang:
log(x \sqrt{x}) = log(x * x^{1/2}) = log(x^{3/2}) = (3/2)log(x)
log(\sqrt{y}) = (1/2)log(y)
log(x/y^2) = log(x) - log(y^2) = log(x) - 2log(y)

Jadi, pembilang menjadi:
(3/2)log(x) - (1/2)log(y) + log(x) - 2log(y)
Gabungkan suku-suku yang sejenis:
((3/2) + 1)log(x) + ((-1/2) - 2)log(y)
(5/2)log(x) + (-5/2)log(y)
(5/2)(log(x) - log(y))
(5/2)log(x/y)

Sekarang, masukkan kembali ke dalam ekspresi awal:
[(5/2)log(x/y)] / log(x/y)

Kita bisa membatalkan log(x/y) selama log(x/y) tidak sama dengan nol (artinya x/y ≠ 1).

Hasilnya adalah 5/2.

Jadi, (log(x \sqrt{x}) - log(\sqrt{y}) + log(x/y^2)) / log(x/y) = 5/2.