Defined in header <complex>  

template< class T > complex<T> log( const complex<T>& z ); 
Computes complex natural (base e) logarithm of a complex value z
with a branch cut along the negative real axis.
z    complex 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.
Errors are reported consistent with math_errhandling.
If the implementation supports IEEE floatingpoint arithmetic,
std::log(std::conj(z)) == std::conj(std::log(z))
z
is (0,+0)
, the result is (∞,π)
and FE_DIVBYZERO
is raised z
is (+0,+0)
, the result is (∞,+0)
and FE_DIVBYZERO
is raised z
is (x,+∞)
(for any finite x), the result is (+∞,π/2)
z
is (x,NaN)
(for any finite x), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (∞,y)
(for any finite positive y), the result is (+∞,π)
z
is (+∞,y)
(for any finite positive y), the result is (+∞,+0)
z
is (∞,+∞)
, the result is (+∞,3π/4)
z
is (+∞,+∞)
, the result is (+∞,π/4)
z
is (±∞,NaN)
, the result is (+∞,NaN)
z
is (NaN,y)
(for any finite y), the result is (NaN,NaN)
and FE_INVALID
may be raised z
is (NaN,+∞)
, the result is (+∞,NaN)
z
is (NaN,NaN)
, the result is (NaN,NaN)
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
The semantics of this function are intended to be consistent with the C function clog
.
The following behaviorchanging defect reports were applied retroactively to previously published C++ standards.
DR  Applied to  Behavior as published  Correct behavior 

LWG 2597  C++98  specification mishandles signed zero imaginary parts  erroneous requirement removed 
#include <iostream> #include <cmath> #include <complex> int main() { std::complex<double> z(0, 1); // // r = 1, θ = pi/2 std::cout << "2*log" << z << " = " << 2.*std::log(z) << '\n'; std::complex<double> z2(sqrt(2)/2, sqrt(2)/2); // r = 1, θ = pi/4 std::cout << "4*log" << z2 << " = " << 4.*std::log(z2) << '\n'; std::complex<double> z3(1, 0); // r = 1, θ = pi std::cout << "log" << z3 << " = " << std::log(z3) << '\n'; std::complex<double> z4(1, 0.0); // the other side of the cut std::cout << "log" << z4 << " (the other side of the cut) = " << std::log(z4) << '\n'; }
Output:
2*log(0,1) = (0,3.14159) 4*log(0.707107,0.707107) = (0,3.14159) log(1,0) = (0,3.14159) log(1,0) (the other side of the cut) = (0,3.14159)
complex common logarithm with the branch cuts along the negative real axis (function template) 

complex base e exponential (function template) 

(C++11)(C++11)  computes natural (base e) logarithm (ln(x)) (function) 
applies the function std::log to each element of valarray (function template) 
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