In Exercise, show that the given nonlinear functions are linear when plotted on semilogarithmic or logarithmic paper.
If experimental data are plotted on logarithmic paper and the points lie on a straight line, it is possible to determine the function (see Exercise 1). The following data come from an experiment to determine the functional relationship between the pressure p and the volume V of a gas undergoing an adiabatic (no heat loss) change. From the graph on logarithmic paper, determine p as a function of V.
V (m3) | 0.100 | 0.500 | 2.00 | 5.00 | 10.0 |
p (kPa) | 20.1 | 2.11 | 0.303 | 0.0840 | 0.0318 |
Exercise 1
In Exercise, show that the given nonlinear functions are linear when plotted on semilogarithmic or logarithmic paper.
A function of the form y = axn is straight when plotted on logarithmic paper, since log y = log a + n log x is in the form of a straight line. The variables are log y and log x; the slope can be found from (log y − log a)/log x = n, and the intercept is a. (To get the slope from the graph, it is necessary to measure vertical and horizontal distances between two points. The log y-intercept is found where log x = 0, and this occurs when x = 1.) Plot y = 3x4 on logarithmic paper to verify this analysis.
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