Log reduction

Template:Short description 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 Template:Math-log reduction means that the concentration of remaining contaminants is only Template:Math 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 Template:Math and Template:Math 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 Template:Math-log reduction is achieved, where

$R=lo{g}_{10}{c}_{\mathrm{b}}-lo{g}_{10}{c}_{\mathrm{a}}=-lo{g}_{10}\left(\frac{c_{\mathrm{a}}}{c_{\mathrm{b}}}\right)$.

For the purpose of presentation, the value of Template:Math 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

$R=-lo{g}_{10}\left(\frac{0.725}{580}\right)=-lo{g}_{10}0.00125\approx 2.903$

Rounded down, Template:Math is 2, so a 2-log reduction is achieved.

Conversely, an Template:Math-log reduction means that a reduction by a factor of Template:Math has been achieved.

Log reduction and percentage reduction

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

$P=100\times \frac{c_{\mathrm{b}}-c_{\mathrm{a}}}{c_{\mathrm{b}}}.$

Example

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

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

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 Template:Math is a whole number, an Template:Math-log reduction corresponds to a percentage reduction with Template:Math leading digits "9" in the percentage (provided that it is at least 10%).

See also

• Decimal reduction time