Two frequently occurring nonlinear functions are thepower law,y5BxA, and the expo- nential function, y5BeAx. These relationships are often presented using graph paper with logarithmic coordinates. Before proceeding, it may be necessary for you to review the mathematical rules for logarithms outlined in Appendix E.
3.4.1 Log Log Plots
When plotted on a linear scale, some data take the form of curves 1, 2, or 3 inFigure 3.9(a), none of which are straight lines. Note that none of these curves intersects either axis except at the origin. If straight-line representation is required, we must transform the data by calculating logarithms; plots of log10 or lny versus log10 or lnx yield straight lines, as shown in Figure 3.9(b). The best straight line through the data can be estimated using a suitable regression analysis, as discussed inSection 3.3.4.
When there are many datum points, calculating the logarithms ofx and y can be time- consuming. An alternative is to use a loglog plot. The raw data, not their logarithms, are plotted directly on loglog graph paper; the resulting graph is as if logarithms were calcu- lated to basee. Graph paper with both axes scaled logarithmically is shown inFigure 3.10;
each axis in this example covers two logarithmic cycles. On loglog plots, the origin (0,0) can never be represented; this is because ln 0 (or log100) is not defined.
y
(a) (b)
x 3 2
1 3
1
2 ln y
ln x
FIGURE 3.9 Equivalent curves on linear and loglog graph paper.
If you are not already familiar with log plots, some practice may be required to get used to the logarithmic scales. Notice that the grid lines on the log-scale axes in Figure 3.10 are not evenly spaced. Within a single log cycle, the grid lines start off wide apart and then become closer and closer. The number at the end of each cycle takes a value 10 times larger than the number at the beginning. For example, on the x-axis ofFigure 3.10, 101is 10 times 1 and 102is 10 times 101; similarly, on the y-axis, 102is 10 times 101and 103is 10 times 102. On thex-axis, 101is midway between 1 and 102. This is because log10101 (51) is midway between log101 (50) and log10102 (52) or, in terms of natural logs, because ln101 (52.3026) is midway between ln1 (50) and ln102 (54.6052).
Similar relationships can be found between 101, 102, and 103 on the y-axis. The first grid line after 1 is 2, the first grid line after 10 is 20, not 11, the first grid line after 100 is 200, and so on. The distance between 1 and 2 is much greater than the distance between 9 and 10; similarly, the distance between 10 and 20 is much greater than the distance between 90 and 100, and so on. On logarithmic scales, the midpoint between 1 and 10 is about 3.16.
A straight line on loglog graph paper corresponds to the equation:
y5BxA ð3:11ị
or
lny5lnB1Alnx ð3:12ị
Inspection ofEq. (3.11)shows that, ifAis positive, y50 whenx50. Therefore, a positive value of A corresponds to either curve 1 or curve 2 passing through the origin of Figure 3.9(a). If Ais negative, when x50, y is infinite; therefore, negative A corresponds to curve 3 inFigure 3.9(a), which is asymptotic to both linear axes.
1 2 5 101
101 2×101 5×101 2×102 5×102 103
102
x y
102
20 50
FIGURE 3.10 Loglog plot.
The values of the parametersAand B can be obtained from a straight line on loglog paper as follows.Amay be calculated in two ways:
1. Ais obtained by reading from the axes the coordinates of two points on the line, (x1,y1) and (x2,y2), and making the calculation:
A5lny22lny1
lnx22lnx1 5lnðy2=y1ị
lnðx2=x1ị ð3:13ị
2. Alternatively, if the loglog graph paper is drawn so that the ordinate and abscissa scales are the same—that is, the distance measured with a ruler for a tenfold change in theyvariable is the same as for a tenfold change in thexvariable—Ais the actual slope of the line.Ais obtained by taking two points (x1,y1) and (x2,y2) on the line and measuring the distances betweeny2andy1and betweenx2andx1with a ruler:
A5 distance betweeny2and y1
distance betweenx2 andx1 ð3:14ị
Note that all points x1, y1, x2, and y2 used in these calculations are points on the line through the data;they are not measured datum points.
OnceAis known,Bis calculated fromEq. (3.12)as follows:
lnB5lny12Alnx1 or lnB5lny22Alnx2 ð3:15ị whereB5e(lnB).Bcan also be determined as the value ofywhenx51.
3.4.2 Semi-Log Plots
When plotted on linear-scale graph paper, some data show exponential rise or decay as illustrated inFigure 3.11(a). Curves 1 and 2 can be transformed into straight lines if log10y or lnyis plotted againstx, as shown inFigure 3.11(b).
An alternative to calculating logarithms is using asemi-log plot, also known as alinearlog plot. As shown inFigure 3.12, the raw data,not their logarithms, are plotted directly on semi-log
1
1
2
2
ln y y
(a) (b)
x x
FIGURE 3.11 Graphic repre- sentation of equivalent curves on linear and semi-log graph paper.
paper; the resulting graph is as if logarithms ofyonly were calculated to base e. Zero cannot be represented on the log-scale axis of semi-log plots. InFigure 3.12, values of the dependent vari- abley were fitted within one logarithmic cycle from 10 to 100; semi-log paper with multiple logarithmic cycles is also available. The features and properties of the log scale used for the y-axis inFigure 3.12are the same as those described inSection 3.4.1for loglog plots.
A straight line on semi-log paper corresponds to the equation:
y5BeAx ð3:16ị
or
lny5lnB1Ax ð3:17ị
Values ofAandBare obtained from the straight line as follows. If two points (x1,y1) and (x2,y2) are located on the line,Ais given by:
A5lny22lny1
x22x1 5lnðy2=y1ị
x22x1 ð3:18ị
Bis the value ofy atx50 (i.e.,Bis the intercept of the line at the ordinate). Alternatively, onceAis known,Bcan be determined as follows:
lnB5lny12Ax1 or lnB5lny22Ax2 ð3:19ị Bis calculated ase(lnB).
EXAMP LE 3.2 CELL GROWTH DATA
Data for cell concentrationxversus timetare plotted on semi-log graph paper. Points (t150.5 h, x153.5 g l21) and (t2515 h,x2510.6 g l21) fall on a straight line passing through the data.
(a) Determine an equation relatingxandt.
(b) What is the value of the specific growth rate for this culture?
0 10 20 30 40 50 60 80 100
10 20 30
y
x
40 50
FIGURE 3.12 Semi-log plot.
Solution
(a) A straight line on semi-log graph paper means thatxandtcan be correlated with the equationx5BeAt.AandBare calculated usingEqs. (3.18) and (3.19):
A5ln 10:62ln 3:5 1520:5 50:076 and
lnB5ln 10:62ð0:076ị ð15ị51:215 or
B53:37
Therefore, the equation for cell concentration as a function of time is:
x53:37e0:076t
This result should be checked by, for example, substitutingt150.5:
BeAt153:37eð0:076ịð0:5ị53:55x1
(b) After comparing the empirical equation obtained forxwithEq. (3.4), the specific growth rate μis 0.076 h21.