2: There must be a better way!

Indeed there must. Scientists often use scientific notation (see BLAST and (Im)probability module) to make explicit the number of digits required to write down a number. There is also an alternative numbering system called a “log” scale, which works for positive numbers only. Here is an example for the number one million:

1000000 = 106 so log (1000000) = log (106) = 6.

Here is an example for the number one millionth:

1/1000000 = 0.000001 = 10-6 so log (0.000001) = log (10-6) = -6.

The pattern evident here is that the log is the exponent when the number is expressed as a power of 10.

These examples involve exact powers of 10 and result in integers for the log scale version of the numbers.  We will generalise for numbers that aren’t simple powers of 10. Large numbers produce positive logs and small numbers (involving only non-zero digits to the right of the decimal point) produce negative logs. Before we go further consider the following: there must also be a number with log that is exactly 0, right? We might consider this the balancing point between big and small numbers. On the log scale it is the number 1:

log (1)= log (100) = 0

The following graphic illustrates numbers arranged according to their place value, from negative log values to positive log values:



SOME FINE PRINT THAT YOU SHOULDN’T SKIP: This module talks about log base 10 which is written log 10 or just log– there are other bases, notably log base 2 (log 2 ) used in computers, and the natural log (loge or ln) often used in any science that deals with growth or decay.