Mean finger volume: Is a GLM with log link function appropriate? I have a model where the volume ($V$) of a finger is normally distributed, with mean $\mu = \beta_0 L^{\beta_1}D^{\beta_2}$ (where $L=$ length, $D=$ diameter and $\beta_i \in \Bbb R$ for $i=0,1,2$) and some variance $\sigma^2$.
I was thinking that if this is a GLM (although I am not sure if we have an exponential family here), a log link function would be appropriate since $\ln (\mu)$ is a linear function of the natural log of the predictors:
$$\ln (\mu) = \ln(\beta_0 L^{\beta_1}D^{\beta_2}) = \ln(\beta_0) + \beta_1\ln(L) + \beta_2\ln(D) $$
However, a Poisson distribution is not ideal since $V, L, D \in \Bbb R$ are not counts.
So, I am not sure if I should proceed with this, or if another link function would be more appropriate. Any suggestions would be appreciated.


I have a model where the volume ($V$) of a finger is normally distributed, with mean $\mu = \beta_0 L^{\beta_1}D^{\beta_2}$

You can rewrite this as
a model where the volume ($V$) of a finger is normally distributed, with mean $$\mu = \exp \left( \beta_0^\prime + \beta_1 L^\prime + \beta_2   D^\prime \right)$$ where $L^\prime = \log(L)$, $D^\prime = \log(D)$ , $\beta_0^\prime = \log(\beta_0)$
So the mean can be expressed as a linear function of the independent variables $\beta_0^\prime + \beta_1 L^\prime + \beta_2   D^\prime $ wrapped inside a non-linear function.
That classifies as a GLM. (provided that the deviation parameter $\sigma$ is independent from $L$ and $D$, ie. constant)

although I am not sure if we have an exponential family here

The normal distribution is in the exponential family.

Instead of the Poisson distribution you can use other distributions. The log link does not restrict this. Jarno's comment shows how you can do it in R glm(V ~ log(L) + log(D), family=gaussian(link="log"))
See also What is the objective function to optimize in glm with gaussian and poisson family?
