Please do not double post.
In order for the function to be continuous at a point, it needs to be defined at that point (which it is at x = 1) and needs to approach the same value from the left as from the right.
So that means in order to be continuous at x = 1, then (do you see why?) Solve for C.
I am checking by finding the derivatives of both by first principles, and checking if the values that those equations produce at x=1 are the same, I am not too certain whether I am doing the right thing there :S. And I am mainly stuck on the derivative of ln(x) by first principles. I can get to: lim h->0 ln(1+h/x)/h