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I have a question concerning, for example, the derivation of the equation for radioactive decay.

You start with the following differential equation $$-\lambda \cdot N=\frac{\mathrm dN}{\mathrm dt}$$

And, after separation of variables and integrating, you obtain $$\int^{t}_{t_0} -\lambda \,\mathrm dt=\int^{N}_{N_0} \frac{1}{N}\,\mathrm dN$$

The right side integrates to $\left[ \ln|N|\right]^{N}_{N_0}$, however the result given in textbooks omits the modulus function. Why can you do that?

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This probably is because we physically cannot have $N<0$. When we are talking about radioactive decay we need $N>0$. Therefore, we can omit the absolute value in the natural logarithm since we are not concerned with negative $N$ values.

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