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·e

The complex logarithm:

 Log z = ln r +

Is defined for negative z.

 

Logarithm of zero

 


See also

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