Finding a function that meets these requirements

Hi everyone! This is my first post, so go easy on me :)

I'm trying to find a smooth function that can replace the intersection of a continuous piecewise function made up of two lines of different slopes, one of which starts at the origin. Right now, I'm trying to find a logarithmic function that goes from the origin (or close to it) and smoothly turns into some line mx+b where m>=1, b>0.

What I've been trying to do is find an intersection of some logarithmic function with the line where the slopes are the same. This would allow me to form a smooth, continuous piecewise function thats an upward curve from 0<x<P, and a straight line from P<=x<inf.

After doing 900 mathematica commands, I'm at a complete loss. Does anyone know if this is even possible?

Thanks everyone,

KodeK