Bacterial population growth mathematics: Which of the following equivalent equations correctly relate initial population (N0), final population (N), and the number of generations (n) during binary fission?

Introduction / Context:
Microbial growth in the exponential phase can be modeled mathematically. For binary fission, each generation doubles the population, leading to simple relationships between the number of generations and population size.

Given Data / Assumptions:

• Binary fission: each generation doubles cell number.
• Use base-10 logarithms where log10(2) ≈ 0.3010.
• We compare algebraically equivalent forms.

Concept / Approach:
The fundamental expression is N = N0 * 2^n. Taking log10 on both sides yields log10(N) − log10(N0) = n * log10(2) ≈ 0.3010 * n. Therefore, options (a), (b), and (c) are mathematically equivalent descriptions of the same growth model; any other form that deviates from this equivalence is incorrect.

Step-by-Step Solution:
Start with binary fission model: N = N0 * 2^n. Take log10: log10(N/N0) = n * log10(2). Insert log10(2) ≈ 0.3010 to get a convenient linear form. Recognize all three expressions describe the same growth relationship.

Verification / Alternative check:
Plot log10(N) versus time; the slope equals log10(2)/g, where g is generation time. This confirms consistency among the formulas.

Why Other Options Are Wrong:

• N = N0 * n^2: Quadratic growth, not doubling; does not model binary fission.

Common Pitfalls:
Mixing natural logs and base-10 logs without converting the constant log(2) appropriately.

Final Answer:
All of these (a–c) are correct and equivalent forms.