Let X be a real normed vector space and f:(0,∞)→X. In this paper, we prove the hyperstability of the logarithmic functional equation fxy−yf(x)=0on Γ of Lebesgue measure zero. More precisely, we prove that if f:(0,∞)→X satisfies ‖fxy−yf(x)‖≤ϕ(x,y)for all (x,y)∈Γ+⊂{(x,y):y>α(x)}[resp.Γ−⊂{(x,y):0<y<α(x)}] of Lebesgue measure zero, where α:(0,∞)→(0,∞) is an arbitrary given function and ϕ:(0,∞)×(0,∞)→[0,∞) satisfies the condition ϕ(x,y)∕y→0 as y→∞ [resp. y→0], then f satisfies the functional equation f(xy)=yf(x) for all x>0 andy>0.