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.

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