4.4.2 Maths Skill: Using Logarithms When Investigating Bacteria

Why Logarithms are Used

• When investigating bacterial growth, numbers can increase rapidly, reaching very large values.
• Using logarithms simplifies these large values, making them easier to interpret and analyse.
• Data is often plotted on a logarithmic scale to visualise patterns and trends clearly.

Understanding Logarithms

• A logarithm is the power to which a number (base) must be raised to produce another number.
• For example, in log10(1000) = 3, 10 is raised to the power of 3 to equal 1000.
• In biology, log10 (common logarithm) is typically used when analysing bacterial populations.

Logarithmic Growth of Bacteria

• Bacteria reproduce by binary fission, causing their numbers to double at regular intervals.
• This results in exponential growth, where population size increases rapidly.
• Logarithms are used to track this growth more efficiently and represent it in a linear form.

Practical Use of Logarithms

1. Plotting Bacterial Growth

• The y-axis is often plotted on a log10 scale, while time is plotted on the x-axis.
• A straight-line graph on a logarithmic scale represents exponential growth.

2. Calculating Growth Rates

• Growth rate = .$(\text{log10 final population size} - \text{log10 initial population size}) / \text{time}$

Key Points About Logarithms in Biology

• Exponential growth: Population size doubles over a fixed period.
• Logarithmic representation: Compresses large ranges of values for better data visualisation.
• Logarithmic scales make it easier to identify growth phases (lag, exponential, and stationary).

Example

If a bacterial culture grows from 1,000 to 1,000,000 cells in 5 hours:

• Initial population size: $\text{log10}(1,000) = 3$
• Final population size: $\text{log10}(1,000,000) = 6$
• Growth rate = .$(6 - 3)/5 = :highlight[0.6 \, \text{log units/hour}]$