All calling sequences return an expression sequence l, q, r such that $a\=b^l\cdot q\+r$, where l, q and r are ordinals, nonnegative integers, or polynomials with positive integer coefficients, and q and r are as small as possible.

The Log(a,b) calling sequence computes the unique ordinal numbers $l$, $q$, and $r$ such that $a\=b^l\cdot q\+r$, $0\prec q\prec b$ and $r\prec b^l$, where $\prec$ is the strict ordering of ordinals.

If $b=0$ or $b=1$, a division by zero error is raised.

The log[b](a) and Log(a,b) calling sequences are equivalent. The log(a) calling sequence is equivalent to Log(a,$\mathrm{\omega }$).

If one of a and b is a parametric ordinal and the logarithm cannot be taken, an error is raised.

The log command overloads the corresponding top-level routine log. The top-level command is still accessible via the :- qualifier, that is, as :-log.

$\mathrm{with}\left(\mathrm{Ordinals}\right)$

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$a≔\mathrm{Ordinal}\left(\left[\left[4\&comma;1\right]\&comma;\left[2\&comma;2\right]\&comma;\left[1\&comma;3\right]\&comma;\left[0\&comma;5\right]\right]\right)$

$a≔{\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$b≔\mathrm{Ordinal}\left(\left[\left[2\&comma;1\right]\&comma;\left[0\&comma;2\right]\right]\right)$

$b≔{\mathbf{\omega }}^2\&plus;2$

$l,q,r≔\mathrm{Log}\left(a\&comma;b\right)$

$l,q,r≔2,1,\mathbf{\omega }\cdot 3\&plus;5$

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

$2,1,\mathbf{\omega }\cdot 3\&plus;5$

$b^l$

${\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;2$

$a=·q+r$

${\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5={\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{LessThan}\left(q\&comma;b\right),\mathrm{LessThan}\left(r\&comma;\right)$

$\mathrm{true},\mathrm{true}$

$l,q,r≔\mathrm{Log}\left(a\&comma;b+1\right)$

$l,q,r≔1,{\mathbf{\omega }}^2\&plus;2,\mathbf{\omega }\cdot 3\&plus;5$

$a={\left(b+1\right)}^l·q+r$

${\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5={\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{LessThan}\left(q\&comma;b+1\right),\mathrm{LessThan}\left(r\&comma;{\left(b+1\right)}^l\right)$

$\mathrm{true},\mathrm{true}$

$\log \left(a\right)$

$4,1,{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Split}\left(a\&comma;\mathrm{degree}=\mathrm{degree}\left(a\right)\right)$

${\mathbf{\omega }}^4\&comma;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;{\mathrm{\omega }}^2+2+x\right)$

Error, (in Ordinals:-Sub) unable to subtract 2+x from 2

$\mathrm{Log}\left(a\&comma;{\mathrm{\omega }}^2+3+x\right)$

$1,{\mathbf{\omega }}^2\&plus;2,\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;{\mathrm{\omega }}^2+2\right)$

$2,1,\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;{\mathrm{\omega }}^2+1\right)$

$2,1,{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;{\mathrm{\omega }}^2\right)$

$2,1,{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;\mathrm{\omega }+1+x\right)$

$3,\mathbf{\omega },{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$\mathrm{Log}\left(a\&comma;\mathrm{\omega }\right)$

$4,1,{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$l,q,r≔\mathrm{Log}\left(a\&comma;x+2\right)$

$l,q,r≔\mathbf{\omega }\cdot 4,1,{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

${\left(x+2\right)}^l$

${\mathbf{\omega }}^4$

$a=·q+r$

${\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5={\mathbf{\omega }}^4\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\mathbf{\omega }\cdot 3\&plus;5$

$l,q,r≔\mathrm{Log}\left(100\&comma;3\right)$

$l,q,r≔4,1,19$

$\mathrm{evalf}\left(\frac{\ln \left(100\right)}{\ln \left(3\right)}\right)$

$4.191806548$

$3^{l}q+r$

$100$

$b≔\mathrm{\omega }·2+3$

$b≔\mathbf{\omega }\cdot 2\&plus;3$

$b^b$

${\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;3}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 3$

$a≔\mathrm{Dec}\left(\right)+x$

$a≔{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;3}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}\&plus;x$

$\mathrm{Log}\left(a\&comma;b\right)$

$\mathbf{\omega }\cdot 2\&plus;2,\mathbf{\omega }\cdot 2\&plus;2,{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;2}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2\&plus;1}\cdot 6\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\cdot 2}\cdot 2\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}\&plus;x$

The Ordinals[Log] and Ordinals[log] commands were introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.