Have a silly question.

In the proof that ln(ax)=ln(a) + ln(x)

we show that ln(ax) and ln(x) have the same derivative, therefore they differ by a constant; ln(a)=ln(x) + C

The book then sets x=1, and says C=ln(a), and the rule is proved. But I get C=ln(a)-ln(x), and that x=1 is a special case.

What's my mistake?