# clogf, clog, clogl

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< c‎ | numeric‎ | complex

Defined in header `<complex.h>` |  |
float complex clogf( float complex z ); | (1) | (since C99)
double complex clog( double complex z ); | (2) | (since C99)
long double complex clogl( long double complex z ); | (3) | (since C99)
Defined in header `<tgmath.h>` |  |
#define log( z ) | (4) | (since C99)

1-3) Computes the complex natural (base-e) logarithm of `z` with branch cut along the negative real axis.
4) Type-generic 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.

### Parameters

z | - | complex argument

### 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.

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point 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` is `-0+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` is `x+∞i` (for any finite x), the result is `+∞+πi/2`
- If `z` is `x+NaNi` (for any finite x), the result is `NaN+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` is `NaN+yi` (for any finite y), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `NaN+∞i`, the result is `+∞+NaNi`
- If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

### 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

### 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(-1-0i) (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(-1-0i) (the other side of the cut) = 0.0-3.141593i

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.7.2 The clog functions (p: 195)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - G.6.3.2 The clog functions (p: 543-544)

  - G.7 Type-generic math <tgmath.h> (p: 545)

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.7.2 The clog functions (p: 176-177)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - G.6.3.2 The clog functions (p: 478-479)

  - G.7 Type-generic math <tgmath.h> (p: 480)

### See also

cexp cexpf cexpl (C99)(C99)(C99) | computes the complex base-e exponential (function)
log logf logl (C99)(C99) | computes natural (base-e) logarithm (ln(x)) (function)

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