Log reduction

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

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

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 10^R has been achieved.

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

${\displaystyle P=100~\times ~{\frac {c_{\mathrm {b} }-c_{\mathrm {a} }}{c_{\mathrm {b} }}}.}$ ^[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%).

• Decimal reduction time