Look at it this way.
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?
Maybe the diagram will help show why x=1 is chosen (for the case of a=2).
Remember, in order to discover the constant C, you must eliminate x.
Setting x=1 causes ln(x) to be zero, eliminating x.
Also, you have a typo in your post
ln(a)=ln(x)+C should read ln(ax)=ln(x)+C.
Hence C=ln(ax)-ln(x) but only "removing" x shows the value of C.
Setting x to 1,