log

(log number) (log number base)

Returns the natural logarithm (base e) of number as a float. With a base the answer is the logarithm of number in that base, which is exactly the quotient (/ (log number) (log base)) -- no special case makes an exact power exact, and none is needed: (log 8 2) is 3.0 and (log 1024 2) is 10.0 from the plain quotient on every backend. Every backend computes it with fdlibm (StrictMath.log on the interpreter and the JVM, the same algorithm on WASM), so the digits agree everywhere. The IEEE edges match everywhere: (log 0.0) is -Infinity. A negative argument leaves the real line and answers the principal logarithm in the complex plane, the way sqrt roots a negative: (log -1) is #C(0.0 3.141592653589793), and (log -100) has real part ln 100 and imaginary part pi. The type of the answer therefore depends on the value, not on how the argument was written -- (let ((x -1d0)) (log x)) is complex on every backend.

Each logarithm takes that escape on its own, so (log -8d0 2d0) answers the plane -- real part 3.0, imaginary part pi / ln 2 -- and a complex number or base is likewise just the quotient of two complex logarithms. The real part is exact there for the same reason (log 8 2) is: a real base makes the complex division divide each part on its own (see /), so the real part is the one rounded quotient ln 8 / ln 2.