#math/calculus The **natural logarithm** $\ln x=\log_{e}x$ is the inverse of $e^{x}$: $ \ln(e^{x})=x,\qquad e^{\ln x}=x\ (x>0),\qquad \ln e=1,\qquad \ln1=0. $ The notation $\log x$ often means the common logarithm $\log_{10}x$, so the base should be made explicit when context is unclear. To differentiate $y=\ln x$, write $e^{y}=x$. Implicit differentiation gives $e^{y}y'=1$, hence $ \boxed{\frac{d}{dx}\ln x=\frac1x}\quad(x>0), \qquad \boxed{\frac{d}{dx}\log_{b}x=\frac{1}{x\ln b}} $ for $b>0$, $b\neq1$, and $x>0$. Useful identities for positive inputs are $\ln(xy)=\ln x+\ln y$, $\ln(x/y)=\ln x-\ln y$, and $\ln(x^{r})=r\ln x$.