I am currently taking an experimental chemistry course where we need to fit data to an equation of the form $y=a\exp(bx)$. They recommend "linearizing" this equation by taking the logarithms of both sides to get $\log(y)=\log(a)+bx$, and then perform a linear regression.
This is all fine, and I can do the regression no problem, but I am wondering about statistical properties of the parameter estimates of $a$ and $b$. Being chemists and not statisticians, most of them don't actually know how to perform a linear regression and just use excel's built in feature, and they also just care about $R^2$ instead of using confidence intervals, hypothesis testing, or some other more suitable statistitical test. I would like to use better statistics but I need to make some assumptions and I'm not sure which ones.
Originally I wanted to include error as $y=a\exp(bx)+\epsilon$, but then I get $\log(y)=\log(a\exp(bx)+\epsilon)$ and I can't write it as a linear regression anymore. Alternatively, I could just assume $\log(y)=\log(a)+bx+\epsilon$ which would then require me to incorporate error as $y=a\exp(bx+\epsilon)$, which seems better. Then I could assume $\epsilon$ is normally distributed and carry on with linear regression by least squares as usual. Is this a valid way of incorporating error to get meaningful statistical properties? If not, is there a better way, potentially using some non-normal distribution for $\epsilon$ or incorporating it differently into the model?