What does 'e' represent in the integral form of exponential growth equation given below?
`N_(t)=N_(0)e^(rt)`

Options

A. Population density at time 't'
B. Population density at time 0
C. Intrinsic rate of natural increase
D. The base of natural logarithm

Answer

To answer the question about what 'e' represents in the integral form of the exponential growth equation \( N_t = N_0 e^{rt} \), we can break down the components of the equation step by step.

### Step-by-Step Solution:

1. **Understanding the Equation**: The equation \( N_t = N_0 e^{rt} \) describes how a population grows over time under ideal conditions. Here, \( N_t \) is the population size at time \( t \), \( N_0 \) is the initial population size, \( r \) is the intrinsic rate of natural increase, and \( t \) is the time.

2. **Identifying 'e'**: In this equation, 'e' is a constant that is approximately equal to 2.71828. It is known as the base of the natural logarithm.

3. **Role of 'e' in Exponential Growth**: The presence of 'e' in the equation indicates that the growth of the population is exponential. This means that as time increases, the population grows at a rate proportional to its current size, leading to rapid increases.

4. **Conclusion**: Therefore, 'e' represents the base of natural logarithms, which is crucial for describing continuous growth processes in mathematics and biology.

### Final Answer:
In the integral form of the exponential growth equation \( N_t = N_0 e^{rt} \), 'e' represents the base of natural logarithms.