104 



BINOMIAL AND NORMAL DISTRIBUTIONS 



Ch. 4 



the corresponding A's matched with their logarithms, the X-scale is 

 what is called a logarithmic scale. Figure 4.53 shows the effect of 

 graphing Y = log X against X when X is on a logarithmic scale. 



Uniform scale, logX: 0.0 

 Corresponding X: 1 



0.1 0.2 



I 

 0.3 

 2 



I 

 0.4 



0.5 

 3 



0.6 

 4 



I 



0.7 



5 



I I I 



0.8 0.9 1.0 

 6 7 8 9 10 



Figure 4.52. Matching of the logarithmic and the arithmetic scales of a 



measurement, X. 



10 100 



X on logarithmic scale 



1000 



Figure 4.53. Graph of Y = log 10 X when X is scaled according to the log 10 X 

 as derived from Figure 4.52. 



As can be seen, the graph is a straight line, and, for any equal dis- 

 tance along the horizontal axis, the Y changes by the same amount. 

 It is noted that the A^-axis falls into parts of equal length: one for 

 numbers from 1 to 10, one for numbers from 10 to 100, and another 

 for X's between 100 and 1000. This corresponds to numbers whose 

 logarithms have characteristics of 0, 1, and 2, respectively. Graph 

 paper with one scale logarithmic and the other arithmetic will be 

 called semi-log paper. When it has three repeated sections along 

 one axis (X-axis in Figure 4.53) it is called three-cycle semi-log 

 paper. The three cycles correspond to any three successive char- 

 acteristics of logarithms, that is, to numbers which fall between any 

 three successive powers of 10. 



Figure 4.54 illustrates the use of semi-log paper to rectify an 

 exponential curve. In this case Y = 2e 3x , but any base for the 

 power could be used. Clearly logio Y = logio 2 + 3X logi e; or 



