14. Logarithms

14.1 Definitions for Logarithms

Option variable: %e_to_numlog

Default value: false

When true, r some rational number, and x some expression, %e^(r*log(x)) will be simplified into x^r . It should be noted that the radcan command also does this transformation, and more complicated transformations of this ilk as well. The logcontract command "contracts" expressions containing log.

Function: li [s] (z)

Represents the polylogarithm function of order s and argument z, defined by the infinite series

inf ==== k \ z Li (z) = > -- s / s ==== k k = 1

li [1] is - log (1 - z). li [2] and li [3] are the dilogarithm and trilogarithm functions, respectively.

When the order is 1, the polylogarithm simplifies to - log (1 - z), which in turn simplifies to a numerical value if z is a real or complex floating point number or the numer evaluation flag is present.

When the order is 2 or 3, the polylogarithm simplifies to a numerical value if z is a real floating point number or the numer evaluation flag is present.

Examples:

(%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]

Function: log (x)

Represents the natural (base e) logarithm of x.

Maxima does not have a built-in function for the base 10 logarithm or other bases. log10(x) := log(x) / log(10) is a useful definition.

Simplification and evaluation of logarithms is governed by several global flags:

logexpand - causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for b integer, always simplifies.) If it is set to false, all of these simplifications will be turned off.

logsimp - if false then no simplification of %e to a power containing log's is done.

lognumer - if true then negative floating point arguments to log will always be converted to their absolute value before the log is taken. If numer is also true, then negative integer arguments to log will also be converted to their absolute value.

lognegint - if true implements the rule log(-n) -> log(n)+%i*%pi for n a positive integer.

%e_to_numlog - when true, r some rational number, and x some expression, %e^(r*log(x)) will be simplified into x^r . It should be noted that the radcan command also does this transformation, and more complicated transformations of this ilk as well. The logcontract command "contracts" expressions containing log.

Option variable: logabs

Default value: false

When doing indefinite integration where logs are generated, e.g. integrate(1/x,x), the answer is given in terms of log(abs(...)) if logabs is true, but in terms of log(...) if logabs is false. For definite integration, the logabs:true setting is used, because here "evaluation" of the indefinite integral at the endpoints is often needed.

Option variable: logarc

Default value: false

If true will cause the inverse circular and hyperbolic functions to be converted into logarithmic form. logarc(exp) will cause this conversion for a particular expression exp without setting the switch or having to re-evaluate the expression with ev.

Option variable: logconcoeffp

Default value: false

Controls which coefficients are contracted when using logcontract. 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 logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then logcontract(1/2*log(x)); will give log(sqrt(x)).

Function: logcontract (expr)

Recursively scans the expression expr, transforming subexpressions of the form a1*log(b1) + a2*log(b2) + c into log(ratsimp(b1^a1 * b2^a2)) + c

(%i1) 2*(a*log(x) + 2*a*log(y))$ (%i2) logcontract(%); 2 4 (%o2) a log(x y )

If you do declare(n,integer); then logcontract(2*a*n*log(x)); gives a*log(x^(2*n)). The coefficients that "contract" in this manner are those such as the 2 and the n here which satisfy featurep(coeff,integer). The user can control which coefficients are contracted by setting the option logconcoeffp to the name of a predicate function of one argument. E.g. if you like to generate SQRTs, you can do logconcoeffp:'logconfun$ logconfun(m):=featurep(m,integer) or ratnump(m)$ . Then logcontract(1/2*log(x)); will give log(sqrt(x)).

Option variable: logexpand

Default value: true

Causes log(a^b) to become b*log(a). If it is set to all, log(a*b) will also simplify to log(a)+log(b). If it is set to super, then log(a/b) will also simplify to log(a)-log(b) for rational numbers a/b, a#1. (log(1/b), for integer b, always simplifies.) If it is set to false, all of these simplifications will be turned off.

Option variable: lognegint

Default value: false

If true implements the rule log(-n) -> log(n)+%i*%pi for n a positive integer.

Option variable: lognumer

Default value: false

If true then negative floating point arguments to log will always be converted to their absolute value before the log is taken. If numer is also true, then negative integer arguments to log will also be converted to their absolute value.

Option variable: logsimp

Default value: true

If false then no simplification of %e to a power containing log's is done.

Function: plog (x)

Represents the principal branch of the complex-valued natural logarithm with -%pi < carg(x) <= +%pi .

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