clogf, clog, clogl
From cppreference.com
Defined in header
<complex.h>


(1)  (since C99)  
(2)  (since C99)  
(3)  (since C99)  
Defined in header
<tgmath.h>


#define log( z )

(4)  (since C99) 
13) Computes the complex natural (basee) logarithm of
z
with branch cut along the negative real axis.
4) Typegeneric macro: If
z
has type long double complex, clogl
is called. if z
has type double complex, clog
is called, if z
has type float complex, clogf
is called. If z
is real or integer, then the macro invokes the corresponding real function (logf, log, logl). If z
is imaginary, the corresponding complex number version is called.
Contents 
[edit] Parameters
z    complex argument 
[edit] Return value
If no errors occur, the complex natural logarithm of z
is returned, in the range of a strip in the interval [−iπ, +iπ] along the imaginary axis and mathematically unbounded along the real axis.
[edit] Error handling and special values
Errors are reported consistent with math_errhandling
If the implementation supports IEEE floatingpoint arithmetic,
 The function is continuous onto the branch cut taking into account the sign of imaginary part
 clog(conj(z)) == conj(clog(z))
 If
z
is0+0i
, the result is∞+πi
and FE_DIVBYZERO is raised  If
z
is+0+0i
, the result is∞+0i
and FE_DIVBYZERO is raised  If
z
isx+∞i
(for any finite x), the result is+∞+πi/2
 If
z
isx+NaNi
(for any finite x), the result isNaN+NaNi
and FE_INVALID may be raised  If
z
is∞+yi
(for any finite positive y), the result is∞+πi
 If
z
is+∞+yi
(for any finite positive y), the result is∞+0i
 If
z
is∞+∞i
, the result is+∞+3πi/4
 If
z
is+∞+∞i
, the result is+∞+πi/4
 If
z
is±∞+NaNi
, the result is+∞+NaNi
 If
z
isNaN+yi
(for any finite y), the result isNaN+NaNi
and FE_INVALID may be raised  If
z
isNaN+∞i
, the result is+∞+NaNi
 If
z
isNaN+NaNi
, the result isNaN+NaNi
[edit] Notes
The natural logarithm of a complex number z with polar coordinate components (r,θ) equals ln r + i(θ+2nπ), with the principal value ln r + iθ
[edit] Example
Run this code
#include <stdio.h> #include <math.h> #include <complex.h> int main(void) { double complex z = clog(I); // r = 1, θ = pi/2 printf("2*log(i) = %.1f%+fi\n", creal(2*z), cimag(2*z)); double complex z2 = clog(sqrt(2)/2 + sqrt(2)/2*I); // r = 1, θ = pi/4 printf("4*log(sqrt(2)/2+sqrt(2)i/2) = %.1f%+fi\n", creal(4*z2), cimag(4*z2)); double complex z3 = clog(1); // r = 1, θ = pi printf("log(1+0i) = %.1f%+fi\n", creal(z3), cimag(z3)); double complex z4 = clog(conj(1)); // or clog(CMPLX(1, 0.0)) in C11 printf("log(10i) (the other side of the cut) = %.1f%+fi\n", creal(z4), cimag(z4)); }
Output:
2*log(i) = 0.0+3.141593i 4*log(sqrt(2)/2+sqrt(2)i/2) = 0.0+3.141593i log(1+0i) = 0.0+3.141593i log(10i) (the other side of the cut) = 0.03.141593i
[edit] See also
(C99)(C99)(C99)

computes the complex basee exponential (function) 
(C99)(C99)

computes natural (basee) logarithm (ln(x)) (function) 
C++ documentation for log
