The natural logarithm, ln, is the logarithm with base $\&ExponentialE;=2.71828$...  For $0<x$ we have $\ln \left(x\right)=y$ <==> $x={\&ExponentialE;}^y$.

For complex-valued expressions x, $\ln \left(x\right)=\ln \left(\left|x\right|\right)+\mathrm{I}\arg \left(x\right)$, where $-\mathrm{\pi }$<argument(x)<=$\mathrm{\pi }$.  Throughout Maple, this computation is taken to be the definition of the principal branch of the logarithm.

The log function is the general logarithm.  For $0<x$ and $0<b$ we have ${\log }_b\left(x\right)=y$<==>$x=b^y$.  log is extended to general complex b and x by ${\log }_b\left(x\right)=\frac{\ln \left(x\right)}{\ln \left(b\right)}$.

The default value of the base b is $\&ExponentialE;$.

You can enter the function log with base b using either the 1-D or 2-D calling sequence.  The base can be entered as an index or as the second argument.  Similarly, $e$ can also be entered as exp(1) in 1-D.  See exp for more about the exponential function.

$\mathrm{log10}\left(x\right)={\log }_{10}\left(x\right)$.

$\mathrm{log2}\left(x\right)={\log }_2\left(x\right)$.

$\ln \left(x\right)={\log }_{\&ExponentialE;}\left(x\right)$.

$\ln \left(1\right)$

$0$

$\mathrm{diff}\left(\ln \left(x\right)\&comma;x\right)$

$\frac{1}{x}$

$\ln \left(3.14+2.71\mathrm{I}\right)$

$1.422562238+0.7120258406\mathrm{I}$

$\ln \left(3+4\mathrm{I}\right)$

$\ln \left(3+4\mathrm{I}\right)$

$\mathrm{evalc}\left(\right)$

$\ln \left(5\right)+\mathrm{I}\arctan \left(\frac{4}{3}\right)$

$\ln \left(10000\right)$

$4\ln \left(10\right)$

$\log \left(10000\right)$

$4\ln \left(10\right)$

$\log \left(\exp \left(3\right)\right)$

$3$

$\mathrm{log10}\left(10000\right)$

$4$

$\log \left[10\right]\left(100\right)$

$2$

$\log \left(16\&comma;8\right)$

$\frac{4}{3}$

$\log \left[\exp \left(1\right)\right]\left(x\right)$

$\ln \left(x\right)$

$\log \left[b\right]\left(x\right)$

$\frac{\ln \left(x\right)}{\ln \left(b\right)}$

$\mathrm{log10}\left(65\right)$

$\frac{\ln \left(65\right)}{\ln \left(10\right)}$

$\mathrm{log2}\left(\exp \left(1\right)\right)$

$\frac{1}{\ln \left(2\right)}$

$\mathrm{evalf}\left(\right)$

$1.442695041$

$\log \left[5\right]\left(5x\right)-\log \left[5\right]\left(x\right)$

$\frac{\ln \left(5x\right)}{\ln \left(5\right)}-\frac{\ln \left(x\right)}{\ln \left(5\right)}$

$\mathrm{simplify}\left(\right)$

$1$

$\mathrm{solve}\left(\log \left[6\right]\left(2y\right)=2\&comma;y\right)$

$18$

$\mathrm{convert}\left(\arcsin \left(x\right)\&comma;\ln \right)$

$\mathrm{-I}\ln \left(\mathrm{I}x+\sqrt{-x^2+1}\right)$

The log2 command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.