What is the formula for exponential growth?

The formula for exponential growth is \( P(t) = P_0 \times e^{rt} \).

In this formula, \( P(t) \) represents the amount of something at time \( t \), \( P_0 \) is the initial amount, \( e \) is the base of the natural logarithm (approximately 2.71828), \( r \) is the growth rate, and \( t \) is the time. Exponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value, leading to growth that accelerates over time.

To break it down, \( P_0 \) is where you start. For example, if you have 100 bacteria at the beginning, \( P_0 \) would be 100. The \( e \) in the formula is a constant that helps model continuous growth. The growth rate \( r \) is usually given as a percentage but is used as a decimal in the formula. For instance, a 5% growth rate would be 0.05. Finally, \( t \) is the time over which the growth occurs, which could be in seconds, minutes, hours, days, or any other unit of time.

Let's say you want to find out how many bacteria you have after 3 hours if they grow at a rate of 5% per hour. If you start with 100 bacteria, you would plug these values into the formula: \( P(3) = 100 \times e^{0.05 \times 3} \). Using a calculator, you would find that \( e^{0.15} \) is approximately 1.1618, so \( P(3) \approx 100 \times 1.1618 = 116.18 \). Therefore, you would have about 116 bacteria after 3 hours.