# Logarithm of Negative Number

What is the logarithm of a negative number?

The logarithmic function

*y* = log_{b}(*x*)

is the inverse
function of the exponential function

*x* =* b*^{y}

Since the base b is positive (b>0), the base b raised to the
power of y must be positive (b^{y}>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.

log_{b}(*x*) is undefined for *x *≤ 0

For example, the base 10 logarithm of -5 is undefined:

log_{10}(-5) is undefined

## Complex logarithm

For complex number z in polar form:

*z* = *r*·*e*^{iθ}

The complex logarithm:

Log *z* = ln *r* + *iθ*

Is defined for negative z.

Logarithm of zero ►

## See also