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Electric potential distribution

Given two cylinders which are concentric, whilst being infinitely long, calculate the electric potential distribution between them.

Two cylinder i, and j, have respective radii r_i=2, r_j=9, with respective potentials V_i=-4, V_j=7

I will start with Laplace's Equation in cylindrical coordinates (r,theta,z).

Considering symmetries;
infinitely long implies we can ignore edge effects, hence we neglect the z term.
Cylindrical so it doesn't depend on theta, neglect theta term.

So we simplify Laplace's equation,to the partial second order differential with respect to r.

I find;
V=C*ln|r|+D, where C and D are constants

Applying boundary conditions to calculate C, and D.
V(2)=-4, V(9)=7

Is that the correct way to go about this??

Any and all feedback will be much appreciated, thanks in advance!!