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<h1 class="chapter"> 14. Logarithms </h1>
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<tr><td align="left" valign="top"><a href="#SEC50">14.1 Definitions for Logarithms</a></td><td> </td><td align="left" valign="top">
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<h2 class="section"> 14.1 Definitions for Logarithms </h2>
<dl>
<dt><u>Option variable:</u> <b>%e_to_numlog</b>
<a name="IDX462"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When <code>true</code>, <code>r</code> some rational number, and
<code>x</code> some expression, <code>%e^(r*log(x))</code> will be simplified into <code>x^r</code> . It
should be noted that the <code>radcan</code> command also does this transformation,
and more complicated transformations of this ilk as well.
The <code>logcontract</code> command "contracts" expressions containing <code>log</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>li</b><i> [<var>s</var>] (<var>z</var>)</i>
<a name="IDX463"></a>
</dt>
<dd><p>Represents the polylogarithm function of order <var>s</var> and argument <var>z</var>,
defined by the infinite series
</p>
<table><tr><td> </td><td><pre class="example"> inf
==== k
\ z
Li (z) = > --
s / s
==== k
k = 1
</pre></td></tr></table>
<p><code>li [1]</code> is <code>- log (1 - z)</code>.
<code>li [2]</code> and <code>li [3]</code> are the dilogarithm and trilogarithm functions, respectively.
</p>
<p>When the order is 1, the polylogarithm simplifies to <code>- log (1 - z)</code>,
which in turn simplifies to a numerical value
if <var>z</var> is a real or complex floating point number or the <code>numer</code> evaluation flag is present.
</p>
<p>When the order is 2 or 3,
the polylogarithm simplifies to a numerical value
if <var>z</var> is a real floating point number
or the <code>numer</code> evaluation flag is present.
</p>
<p>Examples:
</p>
<table><tr><td> </td><td><pre class="example">(%i1) assume (x > 0);
(%o1) [x > 0]
(%i2) integrate ((log (1 - t)) / t, t, 0, x);
(%o2) - li (x)
2
(%i3) li [2] (7);
(%o3) li (7)
2
(%i4) li [2] (7), numer;
(%o4) 1.24827317833392 - 6.113257021832577 %i
(%i5) li [3] (7);
(%o5) li (7)
3
(%i6) li [2] (7), numer;
(%o6) 1.24827317833392 - 6.113257021832577 %i
(%i7) L : makelist (i / 4.0, i, 0, 8);
(%o7) [0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]
(%i8) map (lambda ([x], li [2] (x)), L);
(%o8) [0, .2676526384986274, .5822405249432515,
.9784693966661848, 1.64493407, 2.190177004178597
- .7010261407036192 %i, 2.374395264042415
- 1.273806203464065 %i, 2.448686757245154
- 1.758084846201883 %i, 2.467401098097648
- 2.177586087815347 %i]
(%i9) map (lambda ([x], li [3] (x)), L);
(%o9) [0, .2584613953442624, 0.537213192678042,
.8444258046482203, 1.2020569, 1.642866878950322
- .07821473130035025 %i, 2.060877505514697
- .2582419849982037 %i, 2.433418896388322
- .4919260182322965 %i, 2.762071904015935
- .7546938285978846 %i]
</pre></td></tr></table>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>log</b><i> (<var>x</var>)</i>
<a name="IDX464"></a>
</dt>
<dd><p>Represents the natural (base <em>e</em>) logarithm of <var>x</var>.
</p>
<p>Maxima does not have a built-in function for the base 10 logarithm or other bases.
<code>log10(x) := log(x) / log(10)</code> is a useful definition.
</p>
<p>Simplification and evaluation of logarithms is governed by several global flags:
</p>
<p><code>logexpand</code> - causes <code>log(a^b)</code> to become <code>b*log(a)</code>.
If it is set to <code>all</code>, <code>log(a*b)</code> will also simplify to <code>log(a)+log(b)</code>.
If it is set to <code>super</code>, then <code>log(a/b)</code> will also simplify to <code>log(a)-log(b)</code> for rational
numbers <code>a/b</code>, <code>a#1</code>. (<code>log(1/b)</code>, for <code>b</code> integer, always simplifies.) If
it is set to <code>false</code>, all of these simplifications will be turned off.
</p>
<p><code>logsimp</code> - if <code>false</code> then no simplification of <code>%e</code> to a power
containing <code>log</code>'s is done.
</p>
<p><code>lognumer</code> - if <code>true</code> then negative floating point arguments to
<code>log</code> will always be converted to their absolute value before the <code>log</code> is
taken. If <code>numer</code> is also <code>true</code>, then negative integer arguments to <code>log</code>
will also be converted to their absolute value.
</p>
<p><code>lognegint</code> - if <code>true</code> implements the rule <code>log(-n)</code> ->
<code>log(n)+%i*%pi</code> for <code>n</code> a positive integer.
</p>
<p><code>%e_to_numlog</code> - when <code>true</code>, <code>r</code> some rational number, and
<code>x</code> some expression, <code>%e^(r*log(x))</code> will be simplified into
<code>x^r</code> . It should be noted that the <code>radcan</code> command also
does this transformation, and more complicated transformations of this ilk as well.
The <code>logcontract</code> command "contracts" expressions containing <code>log</code>.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>logabs</b>
<a name="IDX465"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>When doing indefinite integration where
logs are generated, e.g. <code>integrate(1/x,x)</code>, the answer is given in
terms of <code>log(abs(...))</code> if <code>logabs</code> is <code>true</code>, but in terms of <code>log(...)</code> if
<code>logabs</code> is <code>false</code>. For definite integration, the <code>logabs:true</code> setting is
used, because here "evaluation" of the indefinite integral at the
endpoints is often needed.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>logarc</b>
<a name="IDX466"></a>
</dt>
<dt><u>Function:</u> <b>logarc</b><i> (<var>expr</var>)</i>
<a name="IDX467"></a>
</dt>
<dd><p>When the global variable <code>logarc</code> is <code>true</code>,
inverse circular and hyperbolic functions are replaced by
equivalent logarithmic functions.
The default value of <code>logarc</code> is <code>false</code>.
</p>
<p>The function <code>logarc(<var>expr</var>)</code> carries out that replacement for
an expression <var>expr</var>
without setting the global variable <code>logarc</code>.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>logconcoeffp</b>
<a name="IDX468"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>Controls which coefficients are
contracted when using <code>logcontract</code>. It may be set to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do <code>logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$</code> . Then
<code>logcontract(1/2*log(x));</code> will give <code>log(sqrt(x))</code>.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>logcontract</b><i> (<var>expr</var>)</i>
<a name="IDX469"></a>
</dt>
<dd><p>Recursively scans the expression <var>expr</var>, transforming
subexpressions of the form <code>a1*log(b1) + a2*log(b2) + c</code> into
<code>log(ratsimp(b1^a1 * b2^a2)) + c</code>
</p>
<table><tr><td> </td><td><pre class="example">(%i1) 2*(a*log(x) + 2*a*log(y))$
(%i2) logcontract(%);
2 4
(%o2) a log(x y )
</pre></td></tr></table>
<p>If you do <code>declare(n,integer);</code> then <code>logcontract(2*a*n*log(x));</code> gives
<code>a*log(x^(2*n))</code>. The coefficients that "contract" in this manner are
those such as the 2 and the <code>n</code> here which satisfy
<code>featurep(coeff,integer)</code>. The user can control which coefficients are
contracted by setting the option <code>logconcoeffp</code> to the name of a
predicate function of one argument. E.g. if you like to generate
SQRTs, you can do <code>logconcoeffp:'logconfun$
logconfun(m):=featurep(m,integer) or ratnump(m)$</code> . Then
<code>logcontract(1/2*log(x));</code> will give <code>log(sqrt(x))</code>.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>logexpand</b>
<a name="IDX470"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>Causes <code>log(a^b)</code> to become <code>b*log(a)</code>. If
it is set to <code>all</code>, <code>log(a*b)</code> will also simplify to <code>log(a)+log(b)</code>. If it
is set to <code>super</code>, then <code>log(a/b)</code> will also simplify to <code>log(a)-log(b)</code> for
rational numbers <code>a/b</code>, <code>a#1</code>. (<code>log(1/b)</code>, for integer <code>b</code>, always
simplifies.) If it is set to <code>false</code>, all of these simplifications will
be turned off.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>lognegint</b>
<a name="IDX471"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>If <code>true</code> implements the rule
<code>log(-n)</code> -> <code>log(n)+%i*%pi</code> for <code>n</code> a positive integer.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>lognumer</b>
<a name="IDX472"></a>
</dt>
<dd><p>Default value: <code>false</code>
</p>
<p>If <code>true</code> then negative floating point
arguments to <code>log</code> will always be converted to their absolute value
before the <code>log</code> is taken. If <code>numer</code> is also <code>true</code>, then negative integer
arguments to <code>log</code> will also be converted to their absolute value.
</p>
</dd></dl>
<dl>
<dt><u>Option variable:</u> <b>logsimp</b>
<a name="IDX473"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>If <code>false</code> then no simplification of <code>%e</code> to a
power containing <code>log</code>'s is done.
</p>
</dd></dl>
<dl>
<dt><u>Function:</u> <b>plog</b><i> (<var>x</var>)</i>
<a name="IDX474"></a>
</dt>
<dd><p>Represents the principal branch of the complex-valued natural
logarithm with <code>-%pi</code> < <code>carg(<var>x</var>)</code> <= <code>+%pi</code> .
</p>
</dd></dl>
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