When we hear the word “LOG” without context we may think of a fallen tree with all its limbs removed ready to be cut into lumber, or for some of us older hams we might think of a book where we were once required to keep a record of every transmission we made. In the case of the question in the title the word log is short for “Logarithms.”
While teaching my General Class Amateur Radio class an old bugaboo came back and bit me. When expressing power gain we refer to it in decibels and to calculate decibels we use log and antilog. You can get your General Class license without understanding logs especially today when questions and answers can be memorized but I believe most hams would like to understand what the question and answer means not just 10 is the answer to this question if it is asked. I was able to obtain a First Class Radiotelephone License without understanding logarithms. It was not that I did not want to know it was just I could not find anyone who either could or would explain the concept to me.
When I finally took a course in electronics where logs were explained I was surprised how simple they were but how difficult they are to explain. Over the next few days I am going to use this blog as a forum to try to explain what and whys of logarithms.
The first part of understanding logs is the understanding powers of tens. Most of the readers of this blogs probably already understand the power of tens but because of the connection that is where we will start today.
The powers of ten is a way we use to express large numbers without having to write out a long line of zeros. We might speak of the speed of a radio wave as 300,000,000 meters per second but we can also say it is 3X108 meaning 3 times 10 times 10 eight times or 3 with 8 zeros following; so 100 =1, 101 = 10, 102 = 100, and 103 = 1000.
Maybe an easy way to understand is when speaking of the square of a number it means a number multiplied by itself (2X2=4) and when speaking of the cube of a number it is a number times itself and then times itself again (2X2X2=8). When a formula tells us to square a number is uses a superscript 2 after the number (22=4) and to cube a number the superscript 3 is used (23=8). Knowing this and applying this knowledge to powers of 10 it can be seen that 102=100 and 103=1,000. 101 would naturally follow down to equal 10 and 100 will equal 1.
The numbers considered above are all Integral numbers or whole numbers. The powers of ten also work with non-integral numbers or numbers which are factions of whole numbers. In the figure 2.2 we see that 2 is the integral number and .2 or 2/10 is the non-integral number. Sense 100 is equal to 1 a minus superscript number or power indicator is used to make up the non-integral numbers. 1/10 or .1 would be written 10-1 which shows the number is one number over from the decimal point. 10-2 (pronounced as ten to the minus two) is equal to .01 and down it goes.
The powers of ten numbers make up the Characteristic numbers of logarithms. This is fine if you are going to calculate power increase of 10, 100, 1,000, 10,000 and so on but more needs to be known when it comes to calculating other numbers such as 5, 15, 25, or any other number not included in the previous pattern. So more on using logarithms will be covered tomorrow’s post.
Log 10,000 = log 104 = 4 (that is, the log of 10,000 it 4)
Log 1,000 = log 103 = 3 (that is, the log of 10,000 it 3)
Log 100 = log 102 = 2 (that is, the log of 10,000 it 2)
Log 10 = log 101 = 1 (that is, the log of 10,000 it 1)
Log 1 = log 100 = 0 (that is, the log of 10,000 it 0)
Log .1= log 10-1 = -1 (that is, the log of .1 it -1)
Log .01= log 10-2 = -2 (that is, the log of .01 it -2)
Log .001= log 10-3 = -3 (that is, the log of .001 it -3)
Log .0001= log 10-4 = -4 (that is, the log of .0001 it -4)