# 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 10n 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.[1]

## Mathematical definition

Let cb and ca 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 and R-log reduction is achieved, where

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

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

Example

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

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

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 10R has been achieved.

## Log reduction and percentage reduction

Reduction is often expressed as a percentage. The closer it is to 100%, the better. Letting cb and ca be as before, a reduction by P % is achieved, where

${\displaystyle P=100~\times ~{\frac {c_{\mathrm {b} }-c_{\mathrm {a} }}{c_{\mathrm {b} }}}.}$[2]
Example

Let, as in the earlier example, the concentration of some contaminant be 580 ppm before and 0.725 ppm after treatment. Then

${\displaystyle P~=~100~\times ~{\frac {580-0.725}{580}}~=~100~\times ~0.99875~=~99.875.}$

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

The following table summarizes the most common cases.

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%

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%).