all bodies get brighter if they come closer to the sun. For comets there are always two ways to get brightness:
1) The light source gets brighter. This counts for asteroids too. The dependency is quadratic, half the distance menas 4x the brightness. For comets, only the exponent is used to describe the brightness evolution in comparison of the solar distance. The value is called activity level or heliocentric brightness variation and described by the letter n. Sometimes the 2.5-fold of n i used and called H. I use n.
This first part is n1=2 because of the quadratic (2) dependency.
2) If the comet gets more energy from the sun its come gets larger. Double the energy means double the gas evolution. We have to think about the processes which destroy the outer parts of the coma, but in principle the gas production / coma size is proportional to the energy level, and therewith quadratic too. In reality, this n2 is slightly less than 2 (around 1.5), but this requires an extra discussion.
For dynamic new comets which are the first time close to the sun, that's it, and n=n1+n2=4 (or around 3,5 if we look closer).
For dynamic old comets we have a third mechanism
3) Dynamic old comets are covered by some type of crust which is rich of dust but poor of water, Considerable of gas production requires cracks and cavities in that crust. These are produced by the energy level to. And again, we may conclude that effect=cause and the amount of cracks increases with the same quadratic law, which leads to n2=2. There is a limit in this case: If all of the surface is active or the sublimation cooling is large, the destruction of the crust stops. Then n3 gets around 0.
For dynamic old comets we have first n=n1+n2+n3=6 (5.5) and later n=n1+n2=4 (3.5).
In this case we have n3=13 (!). This is a huge increase of active surface. I did not saw such an amount ago. All the comets I saw had n<10.
added: for comparision I added a heliocentric light curve / prediction curve. Solar distance is given in form of it's logarithm here. I added some real values in red for better understanding. Sorry: The jump slightly up and down