-log(1-x) less than x/(1-x)

Theorem

The following formula holds for $x<1$ and $x\neq 0$: $$-\log(1-x) < \dfrac{x}{1-x},$$ where $\log$ denotes the logarithm.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.1.34$