dimensional analysis - Units inside a logarithm

I have troubles understanding a seemingly simple integral in a physical context. Take a look at $\int_{V_1}^{V_2} \frac{\mathrm{d}V}{V}$ which appears in isothermal expansions (V being the volume of a gas). Now of course the result is $\ln{V_2}-\ln{V_1}$ or using log laws $\ln{\frac{V_2}{V_1}}$.

The first expression would require you to evaluate the logarithm of a unit of volume which to my unterstanding is impossible. The second expression makes sense, though. How can you account for this apparent discrepancy?