log

log(x: array, /) → array

Calculates an implementation-dependent approximation to the natural (base e) logarithm for each element x_i of the input array x.

Note

The natural logarithm of a complex number \(z\) with polar coordinates \((r,\theta)\) equals \(\ln r + (\theta + 2n\pi)j\) with principal value \(\ln r + \theta j\).

Note

For complex floating-point operands, log(conj(x)) must equal conj(log(x)).

Note

By convention, the branch cut of the natural logarithm is the negative real axis \((-\infty, 0)\).

The natural logarithm is a continuous function from above the branch cut, taking into account the sign of the imaginary component.

Accordingly, for complex arguments, the function returns the natural logarithm in the range of a strip in the interval \([-\pi j, +\pi j]\) along the imaginary axis and mathematically unbounded along the real axis.

Note: branch cuts follow C99 and have provisional status (see Branch Cuts).

Parameters:

x (array) – input array. Should have a floating-point data type.

Returns:

out (array) – an array containing the evaluated natural logarithm for each element in x. The returned array must have a floating-point data type determined by Type Promotion Rules.

Notes

Special cases

For real-valued floating-point operands,

  • If x_i is NaN, the result is NaN.

  • If x_i is less than 0, the result is NaN.

  • If x_i is either +0 or -0, the result is -infinity.

  • If x_i is 1, the result is +0.

  • If x_i is +infinity, the result is +infinity.

For complex floating-point operands, let a = real(x_i), b = imag(x_i), and

  • If a is -0 and b is +0, the result is -infinity + πj.

  • If a is +0 and b is +0, the result is -infinity + 0j.

  • If a is a finite number and b is +infinity, the result is +infinity + πj/2.

  • If a is a finite number and b is NaN, the result is NaN + NaN j.

  • If a is -infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + πj.

  • If a is +infinity and b is a positive (i.e., greater than 0) finite number, the result is +infinity + 0j.

  • If a is -infinity and b is +infinity, the result is +infinity + 3πj/4.

  • If a is +infinity and b is +infinity, the result is +infinity + πj/4.

  • If a is either +infinity or -infinity and b is NaN, the result is +infinity + NaN j.

  • If a is NaN and b is a finite number, the result is NaN + NaN j.

  • If a is NaN and b is +infinity, the result is +infinity + NaN j.

  • If a is NaN and b is NaN, the result is NaN + NaN j.

Changed in version 2022.12: Added complex data type support.