Logarithm of Negative Number
What is the logarithm of a negative number?
The logarithmic function
y = logb(x)
is the inverse function of the exponential function
x = by
Since the base b is positive (b>0), the base b raised to the power of y must be positive (by>0) for any real y. So the number x must be positive (x>0).
The real base b logarithm of a negative number is undefined.
logb(x) is undefined for x ≤ 0
For example, the base 10 logarithm of -5 is undefined:
log10(-5) is undefined
Complex logarithm
For complex number z in polar form:
z = r·eiθ
The complex logarithm:
Log z = ln r + iθ
Is defined for negative z.