Logarithmic Coordinate Paper

a specially ruled paper; usually prepared typographically. It is constructed by laying off along the axes of a rectangular coordinate system the common logarithms of the numbers u (on the x-axis) and ν (on the y-axis). Lines parallel to the axes are then drawn through the points (u, v). Semilogarithmic coordinate paper is also used. It is constructed by laying off the numbers u on one axis of a rectangular coordinate system and the common logarithms of numbers ν on the other axis.

Logarithmic coordinate paper and semilogarithmic coordinate paper are used for plotting the graphs of functions, which on the paper acquire a simpler and more intuitive form and, in a number of cases, reduce to lines. Functions given by equations of the form ν = au^b, where a and b are constant coefficients, are depicted by lines on logarithmic coordinate paper, since such equations reduce to the form

y = bx + log a

after taking logarithms and changing to the coordinate system x = log u, y = log v. Similarly, functions given by equations of the form ν = ab^u are depicted on semilogarithmic coordinate paper by lines. This property of logarithmic coordinate paper and semilogarithmic coordinate paper is used when seeking the analytic form of empirical relations. For example, if a series of points with coordinates ui, vi where ui are the values of the argument u and vi, are the corresponding experimental values of a function v, are plotted on logarithmic coordinate paper and found to lie with sufficient accuracy on a line, then we may take the line as the graph of the function ν = f(u), which can therefore be written in the form ν = au^b. The corresponding relation will have the form ν = ab^u for semilogarithmic coordinate paper. The coefficients a and b are found from the diagram.