Using a single function, how can I model a logarithmic-like increase followed by an exponential-like decay?

Question

Using a single function, how can I model a logarithmic-like increase followed by an exponential-like decay? The transition will occur at some critical value, Tcrit. A graph of my experimental data is given below. Tcrit is approximately 475.

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Answer 1

If you allow the sign function, I have a nice solution for you.

Let $\operatorname{sign}(x) = \begin{cases} 1 & x \geq 0 \\ -1 & x < 0 \end{cases}$

Choose $$f(x) = -\operatorname{sign}(x) \left(\alpha - e^{-\beta x + \operatorname{sign}(x)} \right) $$

where $\alpha$ must be chosen so that $\lim_{x\rightarrow 0^+} f(x) = \lim_{x\rightarrow 0^-} f(x)$, and $\beta$ controls the slope.

You can shift and scale from there. Here's what it looks like when $\alpha = 1.55$ and $\beta = 1.5$ (note that this value of $\alpha$ does not satisfy the condition I gave above).