As guest mentioned in a comment, the two expressions are equivalent. Suppose $x > 0$ so the logarithm is defined without any problems.
For logarithms, $\log(2x) = \log (2) + \log (x)$. Consider graphing $x \mapsto \log (x)$ and $x \mapsto \log (2x)$, and maybe their difference, to illustrate. A good window would be $x=1\ldots 10$, $y=0\ldots 5$, as recommended by user52817.
Hence, the first integral equals $\log (x) + \log (2) + C_1$, where $C_1$ is the constant of integration.
The second integral equals $\log (x) + C_2$, where $C_2$ is the constant of integration.
To see that these are the same thing, choose $C_1 = C_2 - \log(2)$.
In particular, as $C_1$ goes through all the real numbers, and as $C_2$ does so, you get precisely the same collections of functions from both integrals.
This would be a nice moment to discuss the meaning of the constant of integration, and of the indefinite integral in general. You might want to see this questions and answers for motivation: Should we avoid indefinite integrals?
This might also be useful: Explaining the symbols in definite and indefinite integrals