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From several different sources, I found two different q-analogies of the Exponential function. According to this and this sources exponential function has two q-analogies given by: $$\exp_q(x)=\sum_{n=0}^{\infty}\dfrac{x^n}{[n]_q!}.\tag1$$ and $$\text{Exp}_q(x)=\sum_{n=0}^{\infty}q^{\frac{n(n-1)}{2}}\dfrac{x^n}{[n]_q!}.\tag2$$

However, the q-derivative of the second is not exactly itself.
So the most natural definition is the first one (for me ) and I would like to know that:

Is there an inverse (logarithmic) function for the q-exponential function defined in $(1)$ ?

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    [here](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&uact=8&ved=0ahUKEwilpO7mqbvRAhWIhlQKHQljCIwQFggdMAE&url=https%3A%2F%2Farxiv.org%2Fpdf%2F0709.1788&usg=AFQjCNHlJdfHJLIljI-QF3ZVoUUTB_Vugw) I found few different definitions of q-logarithms. But no one mention as the inverse of $\exp_q(x).$ – 2017-01-12
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    You can try the [Lagrange inversion theorem](https://en.wikipedia.org/wiki/Lagrange_inversion_theorem) – 2017-01-12

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