x 1 + x ≤ log ( 1 + x ) ≤ x for all − 1 < x {\displaystyle {\frac {x}{1+x}}\leq \log(1+x)\leq x{\mbox{ for all }}-1<x} 2 x 2 + x ≤ x 1 + x + x 2 / 12 ≤ log ( 1 + x ) ≤ x 1 + x ≤ x 2 2 + x 1 + x for 0 ≤ x , reverse for − 1 < x ≤ 0 {\displaystyle {\frac {2x}{2+x}}\leq {\frac {x}{\sqrt {1+x+x^{2}/12}}}\leq \log(1+x)\leq {\frac {x}{\sqrt {1+x}}}\leq {\frac {x}{2}}{\frac {2+x}{1+x}}{\mbox{ for }}0\leq x{\mbox{, reverse for }}-1<x\leq 0}
Both are pretty sharp around x=0, but not for large x.
