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Converting $\ln|x| = h$ to $x = e^h$

In Kreyszig’s Advanced Engineering Mathematics, section on First-Order ODEs, it is written:

$ dF/F = p \, dx $

$ \ln|F| = h = \int p \, dx $

Therefore, $ F = e^h$

However I don’t understand why the absolute value for $F$ is gone.

To my understanding, a simplified version of this could be thought of as $\ln|x| = h$. Then if I write $x = e^h = e^{\ln|x|}$ just like the reasoning of the book, when I put $x = -2$, the equation would not hold true because $e^{\ln|-2|} = e^{\ln2} = 2$, not $-2$.

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