What is radioactivity? State the law of radioactive decay. Show that radioactive decay is exponential in nature.

Radioactivity: Spontaneous decay of naturally occurring unstable nuclei by emission of certain sub-particles (like $\mathrm{\alpha }$,$\mathrm{\beta }$, and $\mathrm{\gamma }$ radiation) is called natural radioactivity.
By radioactivity process of emitting these rays nucleus tries to attain the stability.
Law of radioactivity: According to Rutherford and Soddy, the rate of decay or the number of nuclei decay per unit time at any instant is directly proportional to the number of nuclei (N) present at that instant and is independent of the external physical conditions like temperature, pressure, chemical composition etc.
Let ‘N’ be the number of radioactive atoms present at a time ‘t’ and ‘N₀’ is the initial number of radioactive nuclei. Let ‘dN’ atoms disintegrate in time ‘dt’.
$\therefore$ According to the law of radioactive decay
$\left(\frac{\mathrm{dN}}{\mathrm{dt}}\right)\propto N\Rightarrow \frac{\mathrm{dN}}{\mathrm{dt}}=-\mathrm{\lambda N} ..... \left(1\right)$
The proportionality constant ‘ $\mathrm{\lambda }$ ’is called decay constant or disintegration constant.
The negative sign indicates that as time increases ‘N’ decreases.
From equation (1) $\frac{\mathrm{dN}}{\mathrm{N}}=-\mathrm{\lambda dt} ..... \left(2\right)$
On integration, we get $\int \frac{\mathrm{dN}}{\mathrm{N}}=-\mathrm{\lambda }\int \mathrm{dt}$
${\log }_{\mathrm{e}}^{ }N=-\mathrm{\lambda t}+\mathrm{C} .....\left(3\right)$
Here C is the constant of integration.
$\text{ At }\mathrm{t}=0,\mathrm{N}={\mathrm{N}}_0$, substituting in equation (3),
we get ${\log }_{\mathrm{e}}^{ }{\mathrm{N}}_0=\mathrm{C}$
$\begin{array}{l}\therefore {\log }_{\mathrm{e}}^{ }\mathrm{N}=-\mathrm{\lambda t}+{\log }_{\mathrm{e}}^{ }{\mathrm{N}}_0\\ \Rightarrow {\log }_{\mathrm{e}}^{ }\mathrm{N}-{\log }_{\mathrm{e}}^{ }{\mathrm{N}}_0=-\mathrm{\lambda t}\\ \Rightarrow {\log }_{\mathrm{e}}\left(\frac{\mathrm{N}}{{\mathrm{N}}_0}\right)=-\mathrm{\lambda t};\frac{\mathrm{N}}{{\mathrm{N}}_0}={\mathrm{e}}^{-\mathrm{\lambda t}}\\ \therefore \mathrm{N}={\mathrm{N}}_0{\mathrm{e}}^{-\mathrm{\lambda t}}\dots ..\left(4\right)\end{array}$
This shows that the number of nuclei decreases exponentially with time.