The function powlog(p) returns the formal power series that is the natural logarithm of p.

The power series p must have a nonzero first term ($p\left(0\right)\ne 0$) for its logarithm to be well-defined.

The command with(powseries,powlog) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{powseries}\right)\:$

$t≔\mathrm{powpoly}\left(1+x\,x\right)\:$

$s≔\mathrm{powlog}\left(t\right)\:$

$\mathrm{tpsform}\left(s\,x\,7\right)$

$x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3-\frac{1}{4}{x}^4+\frac{1}{5}{x}^5-\frac{1}{6}{x}^6+O\left(x^7\right)$

$u≔\mathrm{powlog}\left(\mathrm{powdiff}\left(\mathrm{powexp}\left(x\right)\right)\right)\:$

$\mathrm{tpsform}\left(u\,x\,6\right)$

$x+O\left(x^6\right)$

$v≔\mathrm{powlog}\left(\mathrm{powadd}\left(\mathrm{powpoly}\left(1+x\,x\right)\,\mathrm{powexp}\left(x\right)\right)\right)\:$

$\mathrm{tpsform}\left(v\,x\,6\right)$

$\ln \left(2\right)+x-\frac{1}{4}{x}^2+\frac{1}{6}{x}^3-\frac{3}{32}{x}^4+\frac{7}{120}{x}^5+O\left(x^6\right)$