# Name for logarithm variation that works on non-positive values?

I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction.

$$myLog(x) = \begin{cases} \log(x+1), & \mbox{if }x \geq 0 \\ -\log(-x+1), & \mbox{if }x \lt 0 \end{cases}$$

Is there a name for this formula or something similar to it? Has it been studied before? I'd like to read up on it if there is anything out there, but without knowing a name for it I haven't had much success with Google.

Here's a graph of the function. It resembles arctangent, but it doesn't have asymptotes.

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I don't have anything useful to say, other than Hi! How's life been? – Jason DeVito Nov 16 '10 at 15:54
The log function has been extended to complex numbers, perhaps you can consider studying that. en.wikipedia.org/wiki/Complex_logarithm – Aryabhata Nov 16 '10 at 17:25
I doubt this particular formula would have a name; it's simply two logarithmic functions glued together. However, it's quite similar to the inverse hyperbolic sine; in particular, it has the same logarithmic behaviour at infinity. – Rahul Nov 16 '10 at 17:49
Rahul has a good point. The inverse hyperbolic sine has the advantage of being analytic, whereas your function isn't twice differentiable. – Jonas Meyer Nov 16 '10 at 18:35

Consider the inverse of the function you invented, which I'll call $f$. Its inverse is $$f^{-1}(y) = \begin{cases}e^y - 1 & \text{if } y \ge 0, \\ 1 - e^{-y} & \text{if } y \lt 0.\end{cases}$$ Since $e^{-y} \le 1$ when $e \ge 0$, and $e^y < 1$ when $y < 0$, the above inverse function is quite close to the much more nicely-behaved \begin{align}g(y) &= e^y - e^{-y} \\ &\equiv 2 \sinh y.\end{align}
So in some sense, $f(x)$ behaves like $\sinh^{-1} (x/2)$. As it turns out, they agree at $x = 0$ (although their derivatives differ by a factor of two), and as $x \to \infty$, they both approach $\ln x$ and the difference between them decreases as $1/x$.