combine/ln - Maple Help

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combine/ln

combine logarithmic terms

 Calling Sequence combine(f, ln) combine(f, ln, t) combine(f, ln, t, m)

Parameters

 f - any expression t - type m - the name 'symbolic'

Description

 • In the case combine(f, ln), expressions involving sums of logarithms are combined by applying the following transformations:

$a\mathrm{ln}\left(x\right)\to \mathrm{ln}\left({x}^{a}\right)\left(\mathrm{provided}a\cdot \mathrm{argument}\left(x\right)=\mathrm{argument}\left({x}^{a}\right)\right)$

$\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)\to \mathrm{ln}\left(xy\right)\left(\mathrm{provided}\mathrm{argument}\left(xy\right)=\mathrm{argument}\left(x\right)+\mathrm{argument}\left(y\right)\right)$

 where the coefficient $a$ must be a rational constant and the argument of $x$ and $y$ must be in the region where this transformation is valid, unless 'symbolic' is specified.
 • In the case combine(f, ln, t), the first transformation is done if the coefficient $a$ is of type t. Often it is useful to restrict the transformation to be done only if $a$ is an integer instead of a rational. Also, by specifying the type anything, the transformation is done in all valid cases, provided of course that the coefficient $a$ itself is not a logarithm.

Examples

 > $\mathrm{combine}\left(3\mathrm{ln}\left(2\right)-2\mathrm{ln}\left(3\right),\mathrm{ln}\right)$
 ${6}{}{\mathrm{ln}}{}\left(\frac{{1}}{{3}}{}\sqrt{{2}}{}{{3}}^{{2}{/}{3}}\right)$ (1)
 > $\mathrm{combine}\left(a\mathrm{ln}\left(x\right)+3\mathrm{ln}\left(x\right)-\mathrm{ln}\left(1-x\right)+\frac{\mathrm{ln}\left(1+x\right)}{2},\mathrm{ln}\right)$
 ${a}{}{\mathrm{ln}}{}\left({x}\right){+}{3}{}{\mathrm{ln}}{}\left({x}\right){-}{\mathrm{ln}}{}\left({1}{-}{x}\right){+}{\mathrm{ln}}{}\left(\sqrt{{1}{+}{x}}\right)$ (2)
 > $\mathrm{assume}\left(a,\mathrm{real}\right);$$\mathrm{assume}\left(0
 > $\mathrm{combine}\left(a\mathrm{ln}\left(x\right)+3\mathrm{ln}\left(x\right)-\mathrm{ln}\left(1-x\right)+\frac{\mathrm{ln}\left(1+x\right)}{2},\mathrm{ln}\right)$
 ${\mathrm{a~}}{}{\mathrm{ln}}{}\left({\mathrm{x~}}\right){-}{\mathrm{ln}}{}\left({1}{-}{\mathrm{x~}}\right){+}{\mathrm{ln}}{}\left({{\mathrm{x~}}}^{{3}}{}\sqrt{{1}{+}{\mathrm{x~}}}\right)$ (3)
 > $\mathrm{combine}\left(a\mathrm{ln}\left(x\right)+3\mathrm{ln}\left(x\right)-\mathrm{ln}\left(1-x\right)+\frac{\mathrm{ln}\left(1+x\right)}{2},\mathrm{ln},\mathrm{integer}\right)$
 ${-}{\mathrm{ln}}{}\left({-}\frac{{-}{1}{+}{\mathrm{x~}}}{{{\mathrm{x~}}}^{{3}}}\right){+}{\mathrm{a~}}{}{\mathrm{ln}}{}\left({\mathrm{x~}}\right){+}\frac{{1}}{{2}}{}{\mathrm{ln}}{}\left({1}{+}{\mathrm{x~}}\right)$ (4)
 > $\mathrm{combine}\left(a\mathrm{ln}\left(x\right)+3\mathrm{ln}\left(x\right)-\mathrm{ln}\left(1-x\right)+\frac{\mathrm{ln}\left(1+x\right)}{2},\mathrm{ln},\mathrm{anything}\right)$
 ${-}{\mathrm{ln}}{}\left({1}{-}{\mathrm{x~}}\right){+}{\mathrm{ln}}{}\left({{\mathrm{x~}}}^{{\mathrm{a~}}}{}{{\mathrm{x~}}}^{{3}}{}\sqrt{{1}{+}{\mathrm{x~}}}\right)$ (5)
 > $\mathrm{additionally}\left(x,\mathrm{RealRange}\left(0,1\right)\right)$
 > $\mathrm{combine}\left(a\mathrm{ln}\left(x\right)+3\mathrm{ln}\left(x\right)-\mathrm{ln}\left(1-x\right)+\frac{\mathrm{ln}\left(1+x\right)}{2},\mathrm{ln},\mathrm{anything}\right)$
 ${\mathrm{ln}}{}\left(\frac{{{\mathrm{x~}}}^{{\mathrm{a~}}}{}{{\mathrm{x~}}}^{{3}}{}\sqrt{{1}{+}{\mathrm{x~}}}}{{1}{-}{\mathrm{x~}}}\right)$ (6)
 > $\mathrm{combine}\left(b\mathrm{ln}\left(y\right)+3\mathrm{ln}\left(y\right)-\mathrm{ln}\left(1-y\right)+\frac{\mathrm{ln}\left(1+y\right)}{2},\mathrm{ln},\mathrm{anything},\mathrm{symbolic}\right)$
 ${\mathrm{ln}}{}\left(\frac{{{y}}^{{b}}{}{{y}}^{{3}}{}\sqrt{{1}{+}{y}}}{{1}{-}{y}}\right)$ (7)

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