Assuming we have some function with $y = f(a, b, x, e)$, this relationship should be transformed into a simple linear regression function with $y = A + Bx + e$, with $A$ only depending on $a$, $B$ only depending on $b$ and $e$ the normally distributed error-term.
Exercises
(a) We assume that we have a data set with the variables y and x in which y is modeled on $y = a \cdot b^x \cdot exp(e)$. What is $X$ for the transformed (linear) model?
(b) Here we have $y = \exp(\sqrt{a} + \log(b) \cdot \log(x) + e)$. Again we want to we fit with the transformed data set (Y, X) a linear regression. How big is the parameter b?
My approach:
(a) I would use the logarithm, which results in: $log(y) = log(a \cdot b^x \cdot \exp(e)) = \log(a) + x \cdot \log(b) + e$.
Hence x would be: $x = (log(y) - log(a) - e)/log(b)$.
(b) Similar approach: $\log(y) = (\sqrt{a} +log (b) \cdot \log (x) + e)$ and then:
$$\exp((\log(y) - \sqrt{a} - e)/\log(x)) = b$$
Is this approach correct at all?