Logarithms and dilutions

Introduction

Logarithms (log) are often used to simplify calculations with large numbers. When discussing the number of microorganisms in cultures from for instance food samples or when preparing and describing dilutions, logarithms are most often used to express the dilution factor.

Calculations

The logarithm (log) of a certain number (x) is the number (y) to which another number (b) must be raised, to give the number x. That is:

x = b^y

The number b is called the base in the logarithmic system you want to use and most common in science and engineering is that you use the number 10 as a base and then you talk about base-10, decimal or common logarithms, which are usually written log₁₀ or lg₁₀. For the sake of simplicity, only log or lg is often written or said, when referring to the base-10 logarithm. An example:

lg 1000 = 3 because 1000 = 10x10x10, which can also be written 10³.

According to the logarithm rules, e.g. the following expressions apply for all types of logarithms:

log (a x b) = log a + log b
log (a / b) = log a - log b
log a^p = p log a

It follows from the second rule that log₁₀ (1 / b) = 0-log₁₀b = -log₁₀b

I.e. logarithms can also be negative, e.g.:

log₁₀ (1/1000) = -3, since log₁₀ (1/1000) = 0-3 = -3

In other words:
...
10³ = 1000
10² = 100
10¹ = 10
10⁰ = 1 (can not really be explained mathematically, but has for practical reasons been defined as 1) )
10^-1 = 0.1
10^-2 = 0,01
10^-3 = 0.001
...

Dilutions

In microbiology, one often wants to make dilution series when deciding how many microorganisms there are in a sample. The most common is to make a 10-step dilution series to achieve high accuracy at large dilutions. This means that the dilution factor is 1:10 in each step and that from one sample you take 1 part of sample solution to 9 parts of culture medium (or another suitable liquid) in a test tube, mix and transfer 1 part of mixture to 9 parts of culture medium in a new test tube and so on until a suitable degree of dilution has been reached. Then you will get a series of test tubes with dilution factors 1:10, 1: 100, 1: 1000, 1:10 000 and so on. These dilution factors can also be written as: 1/10, 1/100, 1/1000, 1/10 000 and so on, or 0.1, 0.01, 0.001, 0.0001. This can also be expressed with the help of logarithms, i.e. 10^-1, 10^-2, 10^-3, 10^-4 and so on.

Therefore, a 1:1000 dilution is sometimes casually referred to as a minus-three dilution.

Read more about how to use dilution series to determine bacterial amounts under Colony Forming Unit (CFU).

Other logarithms

Logarithms with bases other than 10 are used for practical reasons in different disciplines.

Binary logarithms (lb) with base 2 are used in e.g. computer science, bioinformatics and photography.

Natural logarithms (ln) with base e [a transcendental number (limit value), which is approximately equal to 2.71828 ...] are used in e.g. mathematics, physics and chemistry.