I have data for three exponential curves, I computed log on the $y$ because
$$ y = a e ^{b x} $$
$$ log(y) = log(a eb x) = log(a) + log(eb x) = log(a) + b x. $$
Now from a perfect exponential growth we'd expect to see straight lines. But, for example the blue line, if I have changing slopes, can I interpret that as $b$ also depending on time? Or should I rule out the whole exponential curve as simply a "bad fit"?
Your blue curve indicates two regions where $b$ has two distinct values. This leads to different behavior when you have small or large $x$. I have found that in problems like this the following curve fit can be helpful.
$$\ln\frac ya=\frac{1}{\big(\frac{1}{b_1x}+\frac{1}{b_2x}\bigg)}$$