Knowlpedia

Definition

Natural logarithm

The inverse of the real exponential on the positive real axis.

The natural logarithm is the inverse of the :

log⁡:(0,∞)⟶R,exp⁡(log⁡x)=x.\log:(0,\infty)\longrightarrow\mathbb R, \qquad \exp(\log x)=x.

It is strictly increasing and satisfies

log⁡1=0,log⁡(xy)=log⁡x+log⁡y,ddxlog⁡x=1x.\log 1=0,\qquad \log(xy)=\log x+\log y, \qquad \frac{d}{dx}\log x=\frac1x.

The derivative formula follows from the or by differentiating exp⁡(log⁡x)=x\exp(\log x)=x.

Domain

This real logarithm is defined only for positive inputs. A logarithm of a nonzero complex number requires a branch choice on an appropriate domain; a global single-valued complex logarithm cannot simply be assumed.

References