 Logarithm tangent bound

For the natural logarithm, define $F(u)=u-1-\log u$. Its derivative is $1-1/u$, negative below one and positive above one. Thus its global minimum is $F(1)=0$, proving the bound, with equality only at one. Equivalently $\log (1+x)\le x$ for $x>-1$; this also follows from concavity and the tangent line at one.