Logarithm (${\log }_b$)

The logarithm to base $b>0$, $b\ne 1$, is the inverse of the exponential function $x\mapsto b^x$.

The natural logarithm is the inverse of the real exponential function on the positive real numbers.

For $x>0$, concavity of the natural logarithm givesEquivalently, ${\log }_{b}x\le ({\log }_{b}e)(x-1)$ for $b>1$.

For positive-definite Hermitian operators,The same statement on supports follows by regularizing with a positive multiple of the identity and taking a limit.