Fitting a curve with data that is nonlinear in logs I would like to fit a curve for extrapolation purposes that looks as follows:

The dependent variable is already in logs and the independent variable can be thought of as "days" (i.e. it takes on values 0, 1, 2, 3, 4...). I tried to fit an exponential and a power law model to this data, but the fit was not really good.
I think this is because the data is not linear in logs (at least for the exponential model I am thinking this). Is this correct?
How would you approach such a problem? Are there other types of models that are suitable for data that is non-linear even after taking logs?
Edit: in terms of software, I tried the Exponential and Power law models from the lmfit package (see here, this is a Python package).
Thanks

Seeing that it is a curve fitting regression with just an independent variable that it is easy to visualize, a classic way to deal with non-linearities in the relations is to add new features non-linear in $x$ and then perform a standard linear regression (sometimes this is called linear regression with basis functions).
In this specific case using the lmfit package for me I would use the reciprocal of $x$, i.e $\frac{1}{x}$ and maybe some powers of it too, i.e $\frac{1}{x^2}, \frac{1}{x^2}$ and so on and just use the Model with multiple variables.
Code example, assuming you have an array with $log(y)$ values, let's say ylogged_values and the array of $x$ values called x_values:
from lmfit import Model

def multi_reciprocal_function(x,a0=1, a1=1,a2=1,a3=1):
    return a0 + a1/x + a2/(x**2)+a3*/(x**3)

model = Model(multi_reciprocal_function,independent_vars=['x'])

fit = model.fit(ylogged_values, x = x_values)     

