Why is the slope not back transformed in a regression equation for allometric relationships

Question

I'm learning about allometric relationships and how to derive the parameters from regression equations.

I've seen that you can fit a linear regression model by taking the log of both the X and Y variables of your data that have an allometric relationship.

Then you can use the coefficients of the slope and intercept to create a power law of the form $Y=aX^b$

However, I've read that you need to raise your intercept to whatever base you're using but not your slope. For example, if I got an intercept of -1.2 after conducting a regression on the $log_{10}$ of both X and Y I need to apply $10^{-1.2}$ before using it in my power law.

My question is why don't you have to do this for the slope coefficient?

Answer 1

You have probably found estimates for $p$ and $q$ of the following model:

$$\log_{10}Y = p +q\log_{10}X$$

This is equivalent to

$$10^{\log_{10}Y} = 10^{p +q\log_{10}X} =10^{p +\log_{10}X^q} $$

$$\implies Y = 10^{p}X^q$$

Comparing with your model's original form,

$$Y=aX^b$$

you can see $a=10^p$ and $b=q$, i.e. you only take the power of your slope's estimate.