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Microbial growth · doubling time · logarithmic axes
Exponential growth and logarithmic scales
Calculate populations, generations and doubling times. Read base-ten logarithmic scales and explain why exponential growth can appear as a straight line.
Method and worked examples ↓Learn the method
How to approach these questions
- State the initial population N₀, elapsed time t and doubling time g. Use the same time unit for t and g.
- Calculate the number of doubling intervals n = t/g, then N = N₀ × 2ⁿ. A population model may use non-integer n; it is not a claim that every cell divides together.
- To find time from two populations, calculate n = log(N/N₀)/log(2), using the same logarithm base in numerator and denominator. Then t = ng.
- To estimate doubling time from an exponential interval, use g = t/n. Do not assume a doubling time remains constant outside that interval.
- On a base-ten population axis, equally spaced powers such as 10², 10³ and 10⁴ represent equal tenfold ratios. Zero cannot appear on that logarithmic axis.
Worked examples
A culture starts at 500 cells and follows a constant 20-minute doubling model for 80 minutes. Find its model population.
n = 80/20 = 4. N = 500 × 2⁴ = 8000 cells. This assumes exponential growth throughout the interval.
A model starts at 200 cells and reaches 1200 with doubling time 25 minutes. Find the elapsed time.
n = log(1200/200)/log(2) = log(6)/log(2) ≈ 2.585. t = 25 × 2.585 ≈ 64.6 minutes. The non-integer value refers to a continuous population model.
An interval lasts 90 minutes and population increases from 100 to 800. Estimate doubling time.
n = log(800/100)/log(2) = 3. g = 90/3 = 30 minutes. The estimate depends on the exponential-growth assumption.
A base-ten logarithmic population axis has major ticks 100, 1000 and 10 000. What does an equal vertical interval represent?
Each major interval is a tenfold change, not an increase of 100. A point halfway between 100 and 1000 has value √(100 × 1000) ≈ 316, not 550.
One model, two population scales
The population below starts at 500 and doubles every 20 minutes. It is the same model in both plots. Only the vertical scale changes.
| Time / min | Population / cells |
|---|---|
| 0 | 500 |
| 20 | 1000 |
| 40 | 2000 |
| 60 | 4000 |
| 80 | 8000 |
Where the model stops being useful
A fixed doubling time assumes a constant proportional growth rate. In a closed culture, nutrients can become limiting and waste products can accumulate. Do not extend an exponential-growth calculation through stationary or declining phases unless the question supplies that assumption.
A straight line on these logarithmic population axes supports exponential growth over the displayed interval. A straight line on ordinary linear population axes would describe a constant absolute increase.
Read about populations, energy and nutrients · Practise population estimates from samples
Requirements and model background: AQA mathematical requirements, MS 0.5 and MS 2.5 · OCR A mathematical requirements, M0.5 and M2.5 (6.2.1 and 6.3.2) · OpenStax: population-growth models.
Keep practising
Topic map and study list · Practical data guides · Active recall decks
Course resources
AQA A-level Biology · OCR A A-level Biology · Textbook answers
Skill coverage follows the common A-level science subject content. Check your board’s current requirements: AQA 7402 and OCR A H420.