Question

In the exponential growth equation $N _{ t }= N _{0} e ^{ rt }$, e represents:

• A. The base of number logarithms
• B. The base of exponential logarithms
• C. The base of natural logarithms
• D. The base of geometric logarithms

Answer 1

The Correct Option is C

In the context of the given exponential growth equation, $N_{t} = N_{0} e^{rt}$, let's break down the components to understand the role of e:

1. Explanation of Terms:

   • $N_{t}$: This represents the population size at time t.
   • $N_{0}$: This is the initial population size, or the population size at time t = 0.
   • $r$: The intrinsic rate of increase (growth rate) of the population.
   • $t$: The time period over which the population grows.
   • $e$: A mathematical constant approximately equal to 2.71828, known as Euler's number.

2. The Role of e: In this equation, e serves as the base of the natural logarithms. It is a transcendental number that arises naturally in the study of exponential growth processes, such as population growth, radioactive decay, and compound interest calculations.

3. Clarification of Options:

   • The option "The base of natural logarithms" is correct because e is specifically known as the base of natural logarithms.
   • The option "The base of number logarithms" is incorrect as it does not specify the type of logarithms.
   • The option "The base of exponential logarithms" is misleading because there are no logarithms known as exponential logarithms in standard mathematical terminology.
   • The option "The base of geometric logarithms" is incorrect as geometric logarithms do not exist in conventional mathematics.

4. Conclusion: Therefore, the correct answer is "The base of natural logarithms" as it accurately describes e's role in the equation.