Q.In a certain culture of bacteria the rate of increase is proportional to the number present. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours. How many bacteria were present in the beginning?

Exponential Growth: The Snowball Effect

Imagine you have a single penny that doubles every day. On day one, you have one penny. Day two, two pennies. Day three, four pennies. Day four, eight pennies. By day ten, you have 512 pennies — about five dollars. By day twenty, you have over five thousand dollars. By day thirty, you have over five million dollars.

That is exponential growth. It is not just "fast" growth. It is growth where the rate of increase is proportional to the current amount. The bigger something gets, the faster it grows. This creates a snowball effect that feels slow at first, then suddenly explodes.

The Intuition: What Makes It Different

Compare exponential growth to linear growth. Linear growth adds the same fixed amount each step — like stacking bricks, one per minute. Exponential growth multiplies by a fixed factor each step — like a photocopier that makes a copy of every existing paper, then copies the copies.

The key insight: in exponential growth, the increase itself increases. That is why it eventually outruns any linear process, no matter how fast the linear process starts.

The Precise Mathematical Statement

Exponential growth is described by the differential equation:

where:

• $N$ is the quantity at time $t$
• $r$ is the growth rate (a constant, usually positive)
• $\frac{dN}{dt}$ is the instantaneous rate of change

This equation says: "The speed at which $N$ grows is proportional to $N$ itself." If $N$ doubles, the growth rate doubles. If $N$ triples, the growth rate triples.

The solution to this equation is:

where $N_0$ is the initial quantity at $t=0$, and $e \approx 2.718$ is Euler's number.

This formula gives the quantity at any time $t$, assuming continuous growth at rate $r$.

Doubling Time: A Useful Way to Think

For exponential growth, there is a constant doubling time — the time it takes for the quantity to double. If you start with 100 and it doubles every hour, after one hour you have 200, after two hours 400, after three hours 800. The doubling time does not change, even as the numbers get huge.

The doubling time $T_d$ is related to the growth rate by:

Real-World Examples

Population growth: A bacterial colony that doubles every hour. Start with one bacterium. After 6 hours: 64. After 12 hours: 4096. After 24 hours: over 16 million. …