Linearization of $a(x^b)(e^{cx})$

Question

I'm working on an assignment and I need to solve the equation $y=a\times(x^b)\times(e^{cx})$.

We're given an array of values for $x$ and their corresponding $y$. We need to figure out $a$, $b$, and $c$.

My instructor gave us the hint of linearizing it. I know how to linearize it if the equation was something like $y=a \times e^{bx}$.

I found a similar question on here (below)

Linearization of $axe^{-bx}$

But when I try to calculate it it ends up as $\ln(y) - b\times \ln(x) = cx - \ln(a)$.

We're meant to solve it using graphing tools like Loggerpro or Excel. I don't see how I can graph the left side given the $b$ is mixed in with the information I do have.

Did I do something wrong in the rearranging or am I misunderstanding how linearization works?

Any help would be appreciated.

Answer 1

$y = a x^b e^{cx}$

$log(y) = \log(a) + b \log(x) + cx$

Then, $ z = A + b x_1 + c x_2$, where $z = \log(y), A=\log(a)$, $x_1 = \log(x)$ and $x_2 =x$.

Note that you need to calculate $z$ and $x_1$ based on $x$ and $y$ data points.

This is a simple regression (linear in $A,b,c$) with a standard least squares solution. However, I'm not familiar with graphical tools suggested to you such as Loggerpro.