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.