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## Benford’s Law

Assume that the population in a city grows exponentially at rate \(r\). In other words, the number of people in the city, \(N(t)\), grows as \(N(t)=C e^{rt}\), where \(C<10^6\) is a constant.

1. Determine the time interval \(\Delta t_1\) during which \(N(t)\) will be between 1 and 2 million people.

2. For \(k=1,…,9\), determine the time interval \(\Delta t_k\) during which \(N(t)\) will be between k and k+1 million people.

3. Calculate the total time \(T\) it takes for \(N(t)\) to grow from 1 to 10 million people.

4. Now pick a time \(\hat t \in [0,T]\) uniformly at random, and use the above results to derive the following formula (also known as Benford’s law) $$p_k=\mathbb P(N(\hat t) \in [k, k+1] \,million)=\log_{10}(k+1)-\log_{10}(k).$$