Log reduction

<h2 id="mathematical-definition">Mathematical definition</h2>
<p>Let <i>c</i>b and <i>c</i>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.
</p><p>Then an <i>R</i>-log reduction is achieved, where
</p>

R
        =
        l
        o
        
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            10
          
        
        
          
            c
            
              
                b
              
            
          
        
        −
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        o
        
          g
          
            10
          
        
        
          
            c
            
              
                a
              
            
          
        
        =
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          g
          
            10
          
        
        
          
            (
            
              
                
                  c
                  
                    
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    {\displaystyle R=log_{10}{c_{\mathrm {b} }}-log_{10}{c_{\mathrm {a} }}=-log_{10}{\left({\frac {c_{\mathrm {a} }}{c_{\mathrm {b} }}}\right)}}
  
.
<p>For the purpose of presentation, the value of <i>R</i> is rounded down to a desired precision, usually to a whole number.
</p>
Example

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

R
        =
        −
        l
        o
        
          g
          
            10
          
        
        
          
            (
            
              
                0.725
                580
              
            
            )
          
        
        =
        −
        l
        o
        
          g
          
            10
          
        
        
          0.00125
        
        ≈
        2.903
      
    
    {\displaystyle R=-log_{10}{\left({\frac {0.725}{580}}\right)}=-log_{10}{0.00125}\approx 2.903}

<p>Rounded down, <i>R</i> is 2, so a 2-log reduction is achieved.
</p><p>Conversely, an <i>R</i>-log reduction means that a reduction by a factor of 10<i>R</i> has been achieved.
</p>
<h2 id="log-reduction-and-percentage-reduction">Log reduction and percentage reduction</h2>
<p>Reduction is often expressed as a <a href="/facts/Percentage/Wugt2XEf">percentage</a>. The closer it is to 100%, the better.
Letting <i>c</i>b and <i>c</i>a be as before, a reduction by <i>P</i> % is achieved, where
</p>

P
        =
        100
         
        ×
         
        
          
            
              
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    {\displaystyle P=100~\times ~{\frac {c_{\mathrm {b} }-c_{\mathrm {a} }}{c_{\mathrm {b} }}}.}
  
<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
Example

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

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

<p>So this is (better than) a 99% reduction, but not yet quite a 99.9% reduction.
</p><p>The following table summarizes the most common cases.
</p>
<table><tbody><tr><th>Log reduction</th><th>Percentage</th></tr><tr><td>1-log reduction</td><td>90%</td></tr><tr><td>2-log reduction</td><td>99%</td></tr><tr><td>3-log reduction</td><td>99.9%</td></tr><tr><td>4-log reduction</td><td>99.99%</td></tr><tr><td>5-log reduction</td><td>99.999%</td></tr></tbody></table>
<p>In general, if <i>R</i> is a whole number, an <i>R</i>-log reduction corresponds to a percentage reduction with <i>R</i> leading digits "9" in the percentage (provided that it is at least 10%).
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/D-value_(microbiology)/5HD0rykP">Decimal reduction time</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"Final Report of an NWRI Independent Advisory Panel: Recommended DPR General Guidelines and Operational Requirements for New Mexico" (PDF). National Water Research Institute. January 22, 2016. Retrieved December 7, 2018. <a href="http://www.nwri-usa.org/pdfs/New-Mexico-DPR-Panel-General-Report(1).pdf" target="_blank">http://www.nwri-usa.org/pdfs/New-Mexico-DPR-Panel-General-Report(1).pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"Log and Percent Reductions in Microbiology and Antimicrobial Testing". Microchem Laboratory. December 16, 2015. Retrieved December 7, 2018. <a href="https://microchemlab.com/information/log-and-percent-reductions-microbiology-and-antimicrobial-testing" target="_blank">https://microchemlab.com/information/log-and-percent-reductions-microbiology-and-antimicrobial-testing</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Log reduction open-in-new

Log reduction