Instead of ln(10) you should put ln(base of the logarithm), in this case; ln(e) = 1. When deriving log(x) (base 10), you can rewrite to ln(x)/ln(10). ln(10) is just a constant, so the derivative of this is 1/ln(10) * 1/x.
Also, I think you didn't do the chain rule quite right.
I have the feeling you are guessing more than working systematically. Actually there are two ways to solve this. One is by using the chain rule:
set y = (12x - 5) / (9x - 2). Then the derivative of ln(y) is 1/y . dy/dx.
The other way is to first use ln(a/b) = ln(a) - ln(b) and then use the sum rule to differentiate (you still need the chain rule!).
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