$Q=\ln(p),\; P=p(1-q+\ln(p))$ was given.
I read that one needs to check whether the Poisson brackets $\{Q,P\}=1$ and $\{Q,Q\}=0=\{P,P\}$, but we haven't discussed that kind of method for approval. We did discuss the poisson bracket, and how a transformation is canonical, if a function $K(Q,P,t)$ exists which satisfies both equations:
$\dot{Q_i}=\frac{\partial K}{\partial p_i}$ and $\dot{P_i}=-\frac{\partial K}{\partial Q_i}$ , where $K$ is the Hamiltonian with the new generalised momenta $P$ and new generalised coordinates $Q$. Note that $p,q$ are the old momenta and coordinates. We have also discussed the 4 types of generating functions for a transformation, but I don't know how to solve the exercise with the methods presented in lecture. I don't know how to solve the exercise using these methods. I don't even know how to find out $K$ and all the more the generating function.
I would be very grateful if someone could help me somehow!
Hint: Your transformation may be written as $p=e^Q$, $P=e^Q (1-q+Q)$. This suggests one of the four types of generating functions. Use this to deduce the appropriate generating function and so verify that this is a canonical transformation.