Log reduction

Log reduction is a measure of how thoroughly a decontamination process reduces the concentration of a contaminant. It is defined as the common logarithm of the ratio of the levels of contamination before and after the process, so an increment of 1 corresponds to a reduction in concentration by a factor of 10. In general, an $n$-log reduction means that the concentration of remaining contaminants is only $10^{−n}$ times that of the original. So for example, a 0-log reduction is no reduction at all, while a 1-log reduction corresponds to a reduction of 90 percent from the original concentration, and a 2-log reduction corresponds to a reduction of 99 percent from the original concentration.

Mathematical definition
Let $c_{b}$ and $c_{a}$ be the numerical values of the concentrations of a given contaminant, respectively before and after treatment, following a defined process. It is irrelevant in what units these concentrations are given, provided that both use the same units.

Then an $R$-log reduction is achieved, where


 * $$R=log_{10}{c_\mathrm{b}}-log_{10}{c_\mathrm{a}}=-log_{10}{\left(\frac{c_\mathrm{a}}{c_\mathrm{b}}\right)}$$.

For the purpose of presentation, the value of $R$ is rounded down to a desired precision, usually to a whole number.

Let the concentration of some contaminant be 580 ppm before and 0.725 ppm after treatment. Then
 * Example:


 * $$R=-log_{10}{\left(\frac{0.725}{580}\right)}=-log_{10}{0.00125}\approx 2.903$$

Rounded down, $R$ is 2, so a 2-log reduction is achieved.

Conversely, an $R$-log reduction means that a reduction by a factor of $10^{R}$ has been achieved.

Log reduction and percentage reduction
Reduction is often expressed as a percentage. The closer it is to 100%, the better. Letting $c_{b}$ and $c_{a}$ be as before, a reduction by $P$ % is achieved, where
 * $$P = 100~\times~\frac{c_\mathrm{b} - c_\mathrm{a}}{c_\mathrm{b}}.$$

Let, as in the earlier example, the concentration of some contaminant be 580 ppm before and 0.725 ppm after treatment. Then
 * Example:
 * $$P~=~100~\times~\frac{580 - 0.725}{580}~=~100~\times~0.99875~=~99.875.$$

So this is (better than) a 99% reduction, but not yet quite a 99.9% reduction.

The following table summarizes the most common cases.


 * {| class="wikitable"

! Log reduction ! Percentage
 * 1-log reduction
 * 90%
 * 2-log reduction
 * 99%
 * 3-log reduction
 * 99.9%
 * 4-log reduction
 * 99.99%
 * 5-log reduction
 * 99.999%
 * }
 * 99.99%
 * 5-log reduction
 * 99.999%
 * }
 * }

In general, if $R$ is a whole number, an $R$-log reduction corresponds to a percentage reduction with $R$ leading digits "9" in the percentage (provided that it is at least 10%).