In log-log regression why are the $\beta$1 coefficients the same for non natural logs but the $\beta$0 coefficients different? Can the $\beta$1 coefficients for non natural log-log regression still be interpreted as a 'One percent increase in IV is associated with a $\beta$1 percent increase in DV' Here, it would be for a 1% change in body mass we'd expect to see a 0.6518 % change in brain mass.
body = c(62000,277,5000000,160000,28000,960,200,60000000,37000000,2500)
brain = c(1400,7.5,6000,1700,46.2,7.4,3,6000,7820,12)
fit1 = lm(log(brain)~log(body))
print(fit1)
fit2 = lm(log10(brain)~log10(body))
print(fit2)
fit3 = lm(log2(brain)~log2(body))
print(fit3)
EDIT: Is the maths for the $log_{10}$ case the following (based on Interpretation of log transformed predictor):
In the log-log- model, see that $$\begin{equation*}\beta_1 = \frac{\partial \log_{10}(y)}{\partial \log_{10}(x)}.\end{equation*}$$ which gives $$\begin{equation*} \frac{\partial \log_{10}(y)}{\partial y} = \frac{1}{y} \end{equation*}$$ or $$\begin{equation*} \partial \log_{10}(y) = \frac{\partial y}{y}. \end{equation*}$$