*I used the search, but for some reason I can not find such a question. This is probably very simple, but not for me...*

I want to understand why $\int \frac{1}{x}dx = \ln{x} +\rm const$?

I'm trying to solve this as $$\int \frac{1}{x}dx = \int x^{-1} dx = \int x^{-1} dx = \frac{x^{-1+1}}{-1+1} + {\rm const}=\frac{x^0}{0} + \rm const=\frac{1}{0} + const$$

How do I uncover uncertainty?

I don't know how to apply knowledge about the logarithm ($x=\log\_{a} b \Leftrightarrow a^x=b$) in this case.