Differential equations logarithmus rule question

Question

I'm trying to understand a exercise about differential equations
$x'=\frac{1}{2}x+1$
I'm going for the general solution by using separable equation. Everything goes well until I get off the rails:
$\int\frac{1}{\frac{1}{2}x+1}=2*log(\frac{x+2}{2})+C$
I checked the step with Wolfram Alpha step by step solution and they're doing a step a absolutly can't understand and they come up with this:
$y_{(x)}= 2*log(x+2)+C$
What happend there? Why does the fraction inside the logarithm disappear? What rule was there applied?

Answer 1

Both solutions are correct, because:

$$2\log(\frac{x+1}{2})=2\log(x+1)-2\log 2$$