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 geometric logarithms
• D. The base of natural logarithms

Answer 1

The correct option is D

The base of natural logarithms

The correct option is D.

Explanation of correct option:

1. The exponential growth formula is used to express a function of exponential growth, i.e., when the growth rate of the value of a mathematical function is proportional to the function’s current value, resulting in its growth with time being an exponential function.
2. The equation is: $N_t=N_0{e}^{rt}$
3. Here
   • t = time (number of periods)
   • N_t = population density after time t
   • N₀ = population density at initial time (t = 0)
   • r = Intrinsic rate of natural increase called biotic potential.
   • e = base of the natural numbers = Euler’s number = 2.71828 (approx.)

Explanation of incorrect options:

Option A

1. Logarithm, the exponent or power to which a base must be raised to yield a given number.
2. Expressed mathematically, x is the logarithm of n to the base b if b^x = n, in which case one writes x = log_b n.
3. For example, 2³ = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log₂ 8

Option B

1. One of the basic properties of the numbers is that they may be expressed in exponential form.
2. Mathematically the exponential form has certain advantages while graphically the logarithmic form is more informative.
3. Thus,$\mathrm{y}={\mathrm{b}}^{\mathrm{x}}$.

Option C

1. A logarithm is an exponent.
2. A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
3. $\mathrm{y}={\mathrm{b}}^{\mathrm{x}}$ exponential form
4. $\mathrm{x}={\log }_{\mathrm{b}}\mathrm{y}$ logarithmic form
5. There's no such type of geometrical form.

Final answer: In the exponential growth equation, e represents the base of natural logarithms.