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A translation on the d-dimensional torus $\mathbb{T}^d$ has the logarithmproperty if the Shrinking Target Property holds for the sequence of balls withradius $n^{-\frac{1}{d}}$. On $\mathbb{T}^1$ every irrational translations hasthis property. In higher dimension, we give criterions based upon an unusualDiophantine type. In the two dimensional torus, we construct counter examplesto the logarithm with translations vectors of arbitrary small Diophantine typeand example of logarithm property with Liouvillian vector

Cibles r\'etr\'ecissantes de rayon $n^{-\frac{1}{d}}$ : propri\'et\'e du logarithme