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+ | ====== Function LOG ====== | ||

+ | |||

+ | ====Syntax==== | ||

+ | * **log** //number ''&optional'' base// → //logarithm// | ||

+ | |||

+ | ====Arguments and Values==== | ||

+ | * //number// - a non-zero //[[CL:Glossary:number]]//. | ||

+ | * //base// - a //[[CL:Glossary:number]]//. | ||

+ | * //logarithm// - a //[[CL:Glossary:number]]//. | ||

+ | |||

+ | ====Description==== | ||

+ | **log** returns the logarithm of //number// in base //base//. If //base// is not supplied its value is ''e'', the base of the natural logarithms. | ||

+ | |||

+ | **log** may return a //[[CL:Glossary:complex]]// when given a //[[CL:Glossary:real]]// negative //number//. | ||

+ | |||

+ | <blockquote> | ||

+ | (log -1.0) ≡ (complex 0.0 (float pi 0.0)) | ||

+ | </blockquote> | ||

+ | |||

+ | If //base// is zero, **log** returns zero. | ||

+ | |||

+ | The result of ''(log 8 2)'' may be either ''3'' or ''3.0'', depending on the implementation. An implementation can use floating-point calculations even if an exact integer result is possible. | ||

+ | |||

+ | The branch cut for the logarithm function of one argument (natural logarithm) lies along the negative real axis, continuous with quadrant II. The domain excludes the origin. | ||

+ | |||

+ | The mathematical definition of a complex logarithm is as follows, whether or not minus zero is supported by the implementation: | ||

+ | |||

+ | <blockquote> | ||

+ | (log ''x'') ≡ ([[CL:Functions:complex]] ([[CL:Functions:log]] ([[CL:Functions:abs]] ''x'')) ([[CL:Functions:phase]] ''x'')) | ||

+ | </blockquote> | ||

+ | |||

+ | Therefore the range of the one-argument logarithm function is that strip of the complex plane containing numbers with imaginary parts between ''-π'' (exclusive) and ''π'' (inclusive) if minus zero is not supported, or ''-π'' (inclusive) and~''π'' (inclusive) if minus zero is supported. | ||

+ | |||

+ | The two-argument logarithm function is defined as: | ||

+ | |||

+ | <blockquote> | ||

+ | (log //base// //number//) ≡ ([[CL:Functions:math-divide|/]] (log //number//) (log //base//)) | ||

+ | </blockquote> | ||

+ | |||

+ | This defines the //[[CL:Glossary:principal]]// //[[CL:Glossary:values]]// precisely. The range of the two-argument logarithm function is the entire complex plane. | ||

+ | |||

+ | ====Examples==== | ||

+ | <blockquote> | ||

+ | (log 100 10) <r>2.0 | ||

+ | //or// 2 </r> | ||

+ | (log 100.0 10) <r>2.0 </r> | ||

+ | (log #c(0 1) #c(0 -1)) <r>#C(-1.0 0.0) | ||

+ | //or// #C(-1 0)</r> | ||

+ | (log 8.0 2) <r>3.0</r> | ||

+ | (log #c(-16 16) #c(2 2)) <r>3 | ||

+ | //or// #c(3.0 0.0) ; approximately | ||

+ | //or// 3.0 ; approximately, unlikely</r> | ||

+ | </blockquote> | ||

+ | |||

+ | ====Affected By==== | ||

+ | The implementation. | ||

+ | |||

+ | ====Exceptional Situations==== | ||

+ | None. | ||

+ | |||

+ | ====See Also==== | ||

+ | * **[[CL:Functions:exp|Function EXP]]** | ||

+ | * **[[CL:Functions:expt|Function EXPT]]** | ||

+ | * {\secref\FloatSubstitutability} | ||

+ | |||

+ | ====Notes==== | ||

+ | None. | ||

+ | |||

+ | \issue{REAL-NUMBER-TYPE:X3J13-MAR-89} \issue{IEEE-ATAN-BRANCH-CUT:SPLIT} \issue{IEEE-ATAN-BRANCH-CUT:SPLIT} \issue{COMPLEX-RATIONAL-RESULT:EXTEND} | ||