We start by proving that the gamma distribution is a member of the exponential family. It looks a bit different if either $\alpha$ or $\beta$ are known but below I showcase my proof if both are unknown
$$ pdf: p(x|\beta, \alpha) = \frac{\beta^\alpha}{\Gamma (\alpha)} x^{\alpha-1} e^{-\beta x} $$ $$ log [p(x|\beta, \alpha)] = \alpha log \beta - log\Gamma (\alpha) + (\alpha - 1)logx - \beta x $$ $$ p(x|\beta, \alpha) = \frac{1}{\Gamma (\alpha) }exp [\alpha log \beta + (\alpha - 1)logx - \beta x] $$
If we define a member of the exponential family as
$$ p_(x|\theta) = \frac{h(x)}{z(\theta)} exp(\eta(\theta)^T \cdot s(x)) \quad \quad \quad (1) $$
Then we can see that the gamma distribution is a member of the exponential family, since
$$ h(x)=1, \quad z(\alpha, \beta) = \Gamma(\alpha), \quad \eta_1(\alpha, \beta)= \alpha log \beta, \quad s_1 (x) = 1 \\ \eta_2(\alpha, \beta)=(\alpha-1), \quad s_2(x)=logx, \quad \eta_3(\alpha, \beta) = -\beta, \quad s_3(x)=x $$
My question: Now, we are to find the natural parameter and describe the natural parameter space. Any help here would be greatly appreciated, since I have a hard time understanding the concept of natural parameters, how to calculate it and why we are interested in it.
EDIT: I've tried reading up a bit, and my guess at the moment is that when I successfully write a p.d.f in terms of equation (1), then I can determine that the corresponding distribution is a member of the exponential family, and the function $\eta(\theta)$ is the natural parameter.