In the equation `N_(t)=N_(0)e^(rt),'e'` is

Options

A. Population density after the time 'f'
B. Population density at the time zero
C. Intrinsic rate of natural increase
D. The base of natural logarithm

Answer

To solve the question regarding the equation \( N_t = N_0 e^{rt} \) and to identify what \( e \) represents, we can break it down step by step:

### Step-by-Step Solution:

1. **Understand the Equation**: The equation \( N_t = N_0 e^{rt} \) is used to describe the exponential growth of a population over time. Here, \( N_t \) is the population at time \( t \), \( N_0 \) is the initial population (at time zero), \( r \) is the intrinsic rate of natural increase, and \( t \) is the time.

2. **Identify the Components**:
- \( N_t \): Population density after time \( t \).
- \( N_0 \): Population density at time zero.
- \( r \): Intrinsic rate of natural increase.
- \( t \): Time.
- \( e \): This is the mathematical constant approximately equal to 2.71828.

3. **Focus on the Constant \( e \)**: In the context of this equation, \( e \) serves as the base of the natural logarithm. It is a fundamental constant in mathematics that arises frequently in growth processes, particularly in situations involving continuous growth or decay.

4. **Conclusion**: Since the question specifically asks what \( e \) is in the equation, we can conclude that \( e \) is the base of the natural logarithm.

### Final Answer:
In the equation \( N_t = N_0 e^{rt} \), \( e \) is the base of the natural logarithm.

---