Logarithms and logarithmic graphs

Why use logs?

• log₁₀ x is the power to which 10 is raised to give x: log₁₀ 1000 = 3, log₁₀ (2.5 x 10⁶) = 6.40.
• Quantities spanning several orders of magnitude fit on one graph using log₁₀.
• Exponential growth (each cell divides into two) gives a curve on a normal scale but a straight line on a log scale during the log (exponential) phase.
• To go back: number = 10^{log value}.

Plotting

Two ways:

1. Calculate log₁₀ of each value and plot on ordinary graph paper (axis label: 'log₁₀ (number of cells per cm³)').

2. Use semi-log paper, where one axis is already logarithmic (each cycle is a factor of 10).

On a log axis, equal distances mean equal ratios, not equal differences.

CP13: number of yeast or bacterial cells (from a colorimeter, haemocytometer or plate counts) against time: lag, log, stationary and death phases. The growth rate constant from the log phase is taught in The bacterial growth curve (Unit 4).

Number of generations and doubling

If the population doubles each generation: N = N₀ x 2ⁿ, so n = (log₁₀ N - log₁₀ N₀) / log₁₀ 2.

Mean generation time = time ÷ number of generations.

Worked example

A culture rises from 2.0 x 10³ to 6.4 x 10⁴ cells cm⁻³ in 150 minutes. Calculate the number of generations and the mean generation time.

1. n = (log 64000 - log 2000) / log 2 = log 32 / log 2 = 5
2. Generation time = 150 / 5 = 30 minutes

Answer: 5 generations; 30 minutes per generation.

Check yourself

1. Calculate log₁₀ of 2.5 x 10⁶ to 2 decimal places.
   Show answer

   6.40. log₁₀ 2.5 + 6 = 0.398 + 6 = 6.40.

2. A graph shows log₁₀ (cells per cm³) = 5.30. Calculate the number of cells per cm³, giving the answer as a x 10⁵ by entering a to 2 significant figures.
   Show answer

   2.0. 10⁵.30 = 199,526, which is 1.995 x 10⁵, or 2.0 x 10⁵ to 2 s.f.

3. A population grows from 500 to 8000 cells by doubling. How many generations is this?
   Show answer

   4. 8000 / 500 = 16 = 2⁴, so 4 generations.

4. A bacterial population passes through 6 generations in 2 hours. Calculate the mean generation time in minutes.
   Show answer

   20 minutes. 120 / 6 = 20 minutes.

5. Why does the log phase of a growth curve appear as a straight line when log₁₀ (number) is plotted?
   • Exponential growth multiplies by a constant factor in equal times, which is a constant increase in log
   • Cells stop dividing
   • Log graphs always show straight lines
   • The number of cells is constant
   Show answer

   Exponential growth multiplies by a constant factor in equal times, which is a constant increase in log. Constant ratio becomes constant difference in logs.

6. Which axis label is correct for a graph of log-transformed bacterial counts?
   • Number of bacteria / log
   • Bacteria per log
   • log₁₀ (number of bacteria per cm³)
   • log / cm³
   Show answer

   log₁₀ (number of bacteria per cm³). Name the transformation and the quantity.

7. On a logarithmic axis, what does each equal step between 10, 100 and 1000 represent?
   • A tenfold increase
   • An increase of 10
   • An increase of 100
   • A doubling
   Show answer

   A tenfold increase. Each cycle is a factor of 10.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Biology specification (XBI11/YBI11, first teaching 2018): Unit 3, Unit 6 and Appendix 6 mathematical skills

Last updated 8 November 2026. Spotted a mistake? Tell us.