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# Logarithms

## Definitions for Logarithms

Function: LOG (X)
  the natural logarithm.

  LOGEXPAND[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 B integer, always simplifies.) If it is set to FALSE, all of these simplifications will be turned off.

  LOGSIMP[TRUE] - if FALSE then no simplification of %E to a power containing LOG's is done.

  LOGNUMER[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.

  LOGNEGINT[FALSE] - if TRUE implements the rule LOG(-n) -> LOG(n)+%i*%pi for n a positive integer.

  %E_TO_NUMLOG[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.

Variable: LOGABS
  default: [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.

Variable: LOGARC
  default: [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 without setting the switch or having to re-evaluate the expression with EV.

Variable: LOGCONCOEFFP
  default:[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 (exp)
  recursively scans an exp, transforming subexpressions of the form a1*LOG(b1) + a2*LOG(b2) + c into LOG(RATSIMP(b1^a1 * b2^a2)) + c

      (C1) 2*(A*LOG(X) + 2*A*LOG(Y))$
      (C2) LOGCONTRACT(%);
                                              2  4
      (D3)                             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[FALSE] 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)).

Variable: LOGEXPAND
  default: [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 B integer, always simplifies.) If it is set to FALSE, all of these simplifications will be turned off.

Variable: LOGNEGINT
  default: [FALSE] - if TRUE implements the rule LOG(-n) -> LOG(n)+%i*%pi for n a positive integer.

Variable: LOGNUMER
  default: [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.

Variable: LOGSIMP
  default: [TRUE] - if FALSE then no simplification of %E to a power containing LOG's is done.

Function: PLOG (X)
  the principal branch of the complex-valued natural logarithm with -%PI < CARG(X) <= +%PI .

Function: POLARFORM (exp)
  returns R*%E^(%I*THETA) where R and THETA are purely real.

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