Semi-log plot

In <a href="/facts/Science/IJoaD7rs">science</a> and <a href="/facts/Engineering/RFVg8H3I">engineering</a>, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a <a href="/facts/Logarithmic_scale/CvIGK5TQ">logarithmic scale</a>, the other on a <a href="/facts/Linear_scale/7L3I3hUz">linear scale</a>. It is useful for data with <a href="/facts/Exponential_curve/m4yEqKJ7">exponential</a> relationships, where one <a href="/facts/Variable_(mathematics)/VbQlrvKz">variable</a> covers a large range of values.
All equations of the form 
 
 
 
 y
 =
 λ
 
 a
 
 γ
 x
 
 
 
 
 {\displaystyle y=\lambda a^{\gamma x}}
 
 form straight lines when plotted semi-logarithmically, since taking logs of both sides gives

log
          
            a
          
        
        ⁡
        y
        =
        γ
        x
        +
        
          log
          
            a
          
        
        ⁡
        λ
        .
      
    
    {\displaystyle \log _{a}y=\gamma x+\log _{a}\lambda .}

This is a line with slope 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
 and 
 
 
 
 
 log
 
 a
 
 
 ⁡
 λ
 
 
 {\displaystyle \log _{a}\lambda }
 
 vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2:

log
        ⁡
        (
        y
        )
        =
        (
        γ
        log
        ⁡
        (
        a
        )
        )
        x
        +
        log
        ⁡
        (
        λ
        )
        .
      
    
    {\displaystyle \log(y)=(\gamma \log(a))x+\log(\lambda ).}

A log–linear (sometimes log–lin) plot has the logarithmic scale on the y-axis, and a <a href="/facts/Linear/fChtg3KC">linear</a> scale on the x-axis; a linear–log (sometimes lin–log) is the opposite. The naming is output–input (y–x), the opposite order from (x, y).
On a semi-log plot the spacing of the scale on the y-axis (or x-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values (or x values) to their log, and plotting the data on linear scales. A <a href="/facts/Log%E2%80%93log_plot/VES46X8k">log–log plot</a> uses the logarithmic scale for both axes, and hence is not a semi-log plot.

Semi-log plot open-in-new

Semi-log plot