If you're free to choose your own values for P, m, and b, then it might be easier to approach the splice from the other direction.

I hope that I understand your exercise correctly.

You could shift a log function so that it passes through the origin.

Determine the slopes of this curve at both the origin and at x = P.

Once you know these slopes, you can create lines using the Point-Slope formula.

If you're not free to choose P, m, and b, then please post all of the given information, and show us a few of the approaches that you tried in mathematica.

Cheers ~ Mark