I reach a point where in the book the author substitutes $\beta A \alpha^{1-\gamma}$ with $c^\gamma$ to simplify the rest of notation, where $\beta, \gamma \in (0,1)$ and $\alpha, A$ two other positive numbers. Is this generally possible, independently of the values of the coefficients?
I can obviously understand the substitution $\beta^\gamma A^\gamma \alpha^{\gamma} = c^\gamma$, but not this.
I would guess that the reasoning is something like: $$ \beta A \alpha^{1-\gamma} = e^{\ln \beta} e^{\ln A} e^{\ln \alpha} e^{\ln \alpha^\gamma} = e^{\ln \beta} e^{\ln A} e^{\ln \alpha} e^{\gamma\ln \alpha} = e^{\ln \beta + \ln A +\ln \alpha} e^{\gamma\ln \alpha} = \delta e^{\gamma\ln \alpha} $$ but then?
You could do the following $$\beta A \alpha^{1-\gamma}=((\beta A \alpha^{1-\gamma})^{1/\gamma})^\gamma=(\beta^{1/\gamma}A^{1/\gamma}\alpha^{(1/\gamma)-1})^\gamma=c^\gamma$$ so that $$c=\beta^{1/\gamma}A^{1/\gamma}\alpha^{(1/\gamma)-1}$$