Trouble understanding the algebraic manipulation here

[colerelm]

So I'm working with Differential equations but my question is more about the steps taken(I have the solution in front of me) to get to the answer. Here's a screenshot of the part of the problem I don't understand:

​< link to objectionable page removed >

After integrating the right side, how do they get "ln(C)" instead of just " + C"? After that, how do they take C and multiply it to the right side instead of just adding C?

 

[HallsofIvy]

An arbitrary constant is an arbitrary constant is an arbitrary constant! It doesn't matter what you call it. Yes, they could have written "\(\displaystyle \int 200kdt= 200kt+ C\)". But the left side is a logarithm: \(\displaystyle ln\left(\frac{P}{200- P}\right)\) so to solve for P you need to take the exponential: \(\displaystyle e^{ln\left(\frac{P}{200- P}\right)}= e^{200kt+ C}\). That gives, of course, \(\displaystyle \frac{P}{200- P}= e^{200kt+ C}= e^{200kt}e^C= C' e^{200kt}\) where I have defined \(\displaystyle C'= e^C\). Since C is an arbitrary constant C' is also (well, strictly speaking e to any power is positive but then we should have written absolute values on the left so the result is the same). The author was trying to avoid using both C and C' by calling the constant of integration ln(C) to begin with.