(Copyright © 2000-2001 --
this item revised 30 Dec 2000)

The logarithm of x to (positive) base b is the number you'd have to use as an exponent on b to get x again. Symbolically, you can define the logarithm like this:

y = log_b(x) <==> b^y = x

But there's no way you can raise a positive number b to a power and get a negative result x; therefore there's no real logarithm for a negative number. For the same reason, there's no number that can serve as a logarithm of 0.

However --

If you move outside the reals and allow complex logarithms, then not only negative numbers but all complex numbers except 0 will have logarithms. Start from the famous identity, due to Euler:

pi*i e = -1

Taking log of both sides, you have

pi*i = ln(-1)

Since log(ab) = log(a) + log(b), the log of any negative number will be the log of the corresponding positive number plus the log of -1:

ln(-x) = pi*i + ln(x) for x > 0

Actually, every complex number has an infinite number of complex logarithms. Since 1 = (-1)², ln[(-1)²] = 2*ln(-1) = 2pi*i is a natural logarithm of 1. Just as you can multiply any number by 1 without changing it, you can add ±2pi*i, or any multiple of ±2pi*i, to any natural logarithm and still have a logarithm of the same number.

- Stan Brown

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