How to get exponential regression equation after performing linear regression on the ln transformed equation?

Question

The equation of an exponential function is $y = ae^{bx}$

Transforming this for linear regression: $\ln(y) = \ln(a) + bx$

Then the linear regression equation is: $\ln(y) = -369.9778 + 0.187693x$

How do I transform it back in the form of $y=ae^{bx}$??

My issue is in $\ln(a) = -369.9778$. Of how to get the $a$ value.

Thank you.

Since I cannot post photos here due to my new joining of the community: Please refer to this post for images and more in-depth analysis:

https://stats.stackexchange.com/questions/324564/how-to-get-exponential-regression-equation-after-performing-linear-regression-on/324568#324568

Answer 1

If $\hat{\beta}_0 = \widehat{\ln \alpha} = -370$, thus $\hat{\alpha} = e^{\hat{\beta}_0} = e^{-370}$. But note that $g(x) = e^x$ is a convex transformation, hence, $\hat{\alpha}$ is a biased estimator of $\alpha$. As such, you may consider using $(\hat{\alpha}, \hat{\beta})$ as initial values for non-linear regression estimation of $y = ae^{bx}$.