 logarithms - Maple Help

simplify/ln

simplify expressions involving logarithms Calling Sequence simplify(expr, ln) Parameters

 expr - any expression ln - literal name; ln Description

 • The simplify/ln function is used to simplify logarithmic expressions. It applies the following simplifications whenever it can determine that the appropriate conditions hold:

 Simplification Provided $\mathrm{ln}\left({a}^{b}\right)$ ==> $b\mathrm{ln}\left(a\right)$ $\mathrm{signum}\left(a\right)=1$ and $b$ is real $\mathrm{ln}\left({a}^{b}\right)$ ==> $b\mathrm{ln}\left(-a\right)$ $\mathrm{signum}\left(a\right)=-1$ and $b$ is even $\mathrm{ln}\left({a}^{b}\right)$ ==> $b\mathrm{ln}\left(a\right)$ $\mathrm{signum}\left(a\right)=-1$ and $b$ is odd $\mathrm{ln}\left({a}^{b}\right)$ ==> $\frac{b\mathrm{ln}\left({a}^{2}\right)}{2}$ $a$ is real and $b$ is even $\mathrm{ln}\left({a}^{b}\right)$ ==> $b\mathrm{ln}\left(a\right)$ $a$ is real and $b$ is odd $\mathrm{ln}\left(xy\right)$ ==> $\mathrm{ln}\left(x\right)+\mathrm{ln}\left(y\right)$ $0 and $\mathrm{signum}\left(y\right)$ is unknown $\mathrm{ln}\left(xy\right)$ ==> $\mathrm{ln}\left(-x\right)+\mathrm{ln}\left(-y\right)$ $\mathrm{signum}\left(x\right)=-1$ $\mathrm{ln}\left(xy\right)$ ==> $\mathrm{ln}\left(-x\right)+\mathrm{ln}\left(y\right)+I\mathrm{\pi }$ $\mathrm{signum}\left(x\right)=-1$ and $\mathrm{signum}\left(y\right)=1$ $\mathrm{ln}\left({ⅇ}^{x}\right)$ ==> $x$ $x$ is real $\mathrm{ln}\left(\mathrm{LambertW}\left(x\right)\right)$ ==> $\mathrm{ln}\left(x\right)-\mathrm{LambertW}\left(x\right)$ $x$ is real

 • In the case of an integer argument to ln, the integer is factored and the logarithm is returned as a sum of logarithms.
 • In the case of a sum of terms as the argument to ln, the integer content of the sum is factored out and the logarithm is returned as a sum of two logarithms.
 • Making the appropriate assumptions on the names in the expression to be simplified (see assume) provides simplify with enough information to apply the above identities correctly. Examples

It is inappropriate to apply the above identities in these cases since nothing is known about n, x, and y:

 > $\mathrm{simplify}\left(\mathrm{ln}\left({x}^{3}\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({{x}}^{{3}}\right)$ (1)
 > $\mathrm{simplify}\left(\mathrm{ln}\left(xy\right)\right)$
 ${\mathrm{ln}}{}\left({x}{}{y}\right)$ (2)
 > $\mathrm{simplify}\left(\mathrm{ln}\left(\mathrm{exp}\left(x\right)\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({{ⅇ}}^{{x}}\right)$ (3)
 > $\mathrm{simplify}\left(\mathrm{ln}\left({y}^{n}\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({{y}}^{{n}}\right)$ (4)

However, by making appropriate assumptions on the variables, enough information is provided to correctly apply the identities:

 > $\mathrm{assume}\left(n,\mathrm{even}\right)$
 > $\mathrm{assume}\left(x,\mathrm{real}\right)$
 > $\mathrm{assume}\left(y<0\right)$
 > $\mathrm{simplify}\left(\mathrm{ln}\left({x}^{3}\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({{\mathrm{x~}}}^{{3}}\right)$ (5)
 > $\mathrm{simplify}\left(\mathrm{ln}\left(xy\right)\right)$
 ${\mathrm{ln}}{}\left({-}{\mathrm{y~}}\right){+}{\mathrm{ln}}{}\left({-}{\mathrm{x~}}\right)$ (6)
 > $\mathrm{simplify}\left(\mathrm{ln}\left(\mathrm{exp}\left(x\right)\right)\right)$
 ${\mathrm{x~}}$ (7)
 > $\mathrm{simplify}\left(\mathrm{ln}\left({y}^{3}\right),\mathrm{ln}\right)$
 ${3}{}{\mathrm{ln}}{}\left({-}{\mathrm{y~}}\right){+}{I}{}{\mathrm{\pi }}$ (8)
 > $\mathrm{simplify}\left(\mathrm{ln}\left({y}^{n}\right),\mathrm{ln}\right)$
 ${\mathrm{n~}}{}{\mathrm{ln}}{}\left({-}{\mathrm{y~}}\right)$ (9)

Simplifications involving integer factors:

 > $\mathrm{simplify}\left(\mathrm{ln}\left(-40a+15b\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({5}\right){+}{\mathrm{ln}}{}\left({-}{8}{}{a}{+}{3}{}{b}\right)$ (10)
 > $\mathrm{simplify}\left(\mathrm{ln}\left(345366\right),\mathrm{ln}\right)$
 ${\mathrm{ln}}{}\left({2}\right){+}{2}{}{\mathrm{ln}}{}\left({3}\right){+}{\mathrm{ln}}{}\left({7}\right){+}{\mathrm{ln}}{}\left({2741}\right)$ (11)