Function log

Syntax:

Arguments and Values:

number—a non-zero number.

base—a number.

logarithm—a number.

Description:

12.5.1 7log returns the logarithm of number in base base. If base is not supplied its value is e, the base of the natural logarithms.

12.5.1 8log may return a complex when given a real negative number.

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

If base is zero, Removed as unnecessary per Barmar since \param{number} is described above as non-zero. and \param{number} is non-zerolog 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.

12.5.3 7The 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.

Per Barmar, this is removed since the above note from IEEE-ATAN-BRANCH-CUT already says the same thing. When \param{number} is a \term{complex}, {\tt log \param{number} \EQ\ (log $\vert$\param{number}$\vert$)+\i{i} \i{phase}(\param{number})\/}. \code (log \param{complex}) \EQ (+ (log (abs \param{complex})) (* (phase \param{complex}) #c(0 1))) \endcode

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

(log x) ≡ (complex (log (abs x)) (phase x))

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

12.5.3 8The two-argument logarithm function is defined as

 (log base number)
 ≡ (/ (log number) (log base))

This defines the principal values precisely. The range of the two-argument logarithm function is the entire complex plane.

Examples:

 (log 100 10)
→ 2.0
→ 2
 (log 100.0 10) → 2.0
 (log #c(0 1) #c(0 -1))
→ #C(-1.0 0.0)
OR→ #C(-1 0)
 (log 8.0 2) → 3.0

!!! Figure out some way to notate these `precise approximations'

 (log #c(-16 16) #c(2 2)) → 3 or approximately #c(3.0 0.0)
                               or approximately 3.0 (unlikely)

Affected By:

The implementation.

Exceptional Situations:

None.

See Also:

exp, expt, Section 12.1.3.3 (Rule of Float Substitutability)

Notes:

None.