From the definition of the decay constant for nuclear decay, derive the exponential decay equation.

By definition, dn/dt = - λ N ,
If you separate the variables which contain N and t, and then integrate w.r.t N and w.r.t t you get,
lnN = -λt + C (1),
To find the constant of integration, solve (1) by setting N(t = 0) = N₀ . This gives a value of,
C = lnN₀ ,
By then substituting the value for C into (1), and relying on the properties of logs and exponentials, the following steps occur,
lnN = -λt + lnN₀ ,
ln(N/N₀) = -λt,
N/N₀ = e^-λt ,
N = N₀e^-λt,