Talk:Order of magnitude/Archive 3

message: The following table contains no useful information.  --ᓛᖁ[ ᑐ] 18:14, 4 Dec 2004 (UTC)
thread_title:  Order of magnitude table 

message: The scale s of the pure double logarithm would have been \log_{10} \log_{10} {x}  as
x > 10: s = \log_{10} \log_{10} x corresponding to x = 10^{10^{s}}, 
but the negative outputs in input range 1 < x < 10 do not have a useful meaning, so we require that x > 10. Furthermore the input minimum is unfortunately only 1, not 0 as needed, so another kind of double-logarithmic scale is added to be used when x < 0.1. 
x < 0.1: s = -\log_{10} (- \log_{10} x) corresponding to x = 10^{-10^{-s}}
These two functions give the value s = 0 when x = 10 and x = 0.1. Because of this a third function is needed that sets s to 0 when the range is 0.1 < x < 10. If it is not desired to collapse this range to a single point, another double-logarithmic scale with offset could be defined by dividing the first function with 10 and multiplying the second with 10. Then both function correspond to the same x = 1 for s = 0.
x > 1: s = \log_{10} (1 + \log_{10} x) corresponding to x = 10^{-1+10^{s}}
x < 1: s = -\log_{10} (1 - \log_{10} x) corresponding to x = 10^{ 1 - 10^{-s} }
 Different types of order of magnitudes
output order magnitude "s"	log10 of	log10 of	log10log10 of	log10(1+log10) of	-log10(1-log10) of	combination with offset
2                         	100          	102          	10100        	1099             	100.99 = 9.77     	1099
1                         	10           	101          	1010         	109              	100.9 = 7.94      	109
0                         	1            	100          	101          	100              	100               	100
-1                        	0.1          	10-1         	100.1 = 1.26 	10-0.9 = 0.126   	10-9              	10-9
-2                        	0.01         	10-2         	100.01 = 1.02	10-0.99 = 0.102  	10-99             	10-99
This introduces an error which is clearly visible for inputs smaller than 1010 and might make the double-logarithmic scale with offset harder to read.
input number x	log10 order of magnitude	combination of log10log10 and -log10(-log10) order of magnitude	offset combination of log10(1+log10) and -log10(1-log10) order of magnitude
10100         	100                          	2                                                              	2.00
1010          	10                           	1                                                              	1.04
102           	2                            	0.30                                                           	0.48
101           	1                            	0                                                              	0.30
100           	0                            	undefined or forced to 0                                       	0
10-1          	-1                           	0                                                              	-0.30
10-2          	-2                           	-0.30                                                          	-0.48
10-10         	-10                          	-1                                                             	-1.04
10-100        	-100                         	-2                                                             	-2.00
Najro 20:21, 8 October 2006 (UTC)
thread_title: Double logarithmic scale