**Measuring noise, part 1**

All of my full time education took place before pocket electronic calculators became readily available to the general public – I graduated from university in 1970 and Sir Clive Sinclair introduced his breakthrough *Executive* model in 1972. So how did we multiply and divide awkward numbers before 1972? We used tables of logarithms or a slide rule; the slide rule was just “logarithms on a stick” and so both methods relied on logarithms.

“So why is the silly old fool telling us this?” I hear you ask. The reason is that the mathematical function known as the logarithm is the basis for the way that noise is measured. Since I intend that this blog will be the first step on a journey that will lead to a thorough grounding in the basic principles of noise measurement, a working knowledge of the logarithmic function seems a good place to start.

Now for the health warning. I did say “mathematical function” and so be prepared for some maths. Now I have been taught a lot of maths in my time, but I have forgotten 99.9% of it and am very shaky on the remaining 0.1%. So I will have to keep the maths really simple, otherwise I will get lost as well as you.

This journey will be worth taking because along the way I will reveal some significant skeletons that I have found in the HS2 closet and explain why it seems that we cannot trust HS2 Ltd on noise.

The logarithmic function was invented by mathematicians as one of the tools that they hoped to use to become masters of the Universe. In the form that we will be using it, where it is called the “common logarithm”, the function is written log_{10}(X), which is pronounced “log of X to the base ten”; other bases can be used and the most common of these is “e” when the function is called the “natural logarithm”, but that one is for real maths geeks. Don’t concern yourself with how to work out the value of log_{10}(X) because you don’t have to if you have a set of logarithmic tables.

After a frantic search I have found and dusted off my copy of *Chambers’s Four Figure Mathematical Tables*, dated 1966, which I purchased to use for my A Levels. I am so proud of this ancient tome that I have included a picture of its cover below. If you aren’t a hoarder like me and haven’t got a set of tables of your own, or if you are too young to have used logs, then a sample table can be found here.

The really useful thing about logarithms can be summed up by the following equation:

log_{10}(A.B) = log_{10}(A) + log_{10}(B), where A.B is A multiplied by B

This means that if you know the values of log_{10}(A) and log_{10}(B) from the log tables, you can add them together to find log_{10}(A.B). You can then use reverse tables, called tables of antilogarithms, to find the value of A.B. So you can find the product A.B by addition alone. How neat is that?

Let’s use the tables and real numbers to work out a simple example; how about 3 times 2? From the tables:

log_{10}(3) = 0.4771 and log_{10}(2) = 0.3010, so the addition of the two logs yields 0.7781. The antilog of 0.7781 is 5.999, so the answer is 6, near enough (well to better than 0.02%, which is due to the tables being limited to four decimal places). Magic!

How about 5 times 2? Again, from the tables:

log_{10}(5) = 0.6990 and log_{10}(2) = 0.3010, so the addition of the two logs yields 1.0000. The antilog of 1.000 is 10, exactly this time. Impressive!

How about 5 times 10?

log_{10}(5) = 0.6990 and log_{10}(10) = 1.0000, so the addition of the two logs yields 1.6990.

This is an interesting result. We know that the answer is 50 and, therefore, that log_{10}(50) is 1.6990. Let’s do one more calculation and then try to work out what is going on. What does 50 times 2 yield?

log_{10}(50) = 1.6990 and log_{10}(2) = 0.3010, so the addition of the two logs yields 2.0000, which must be log_{10}(100).

If we carry on doing this we will find that log_{10}(1,000) is 3.0000, log_{10}(10,000) is 4.0000, etc. So the logarithmic value is cyclic, within increasing powers of ten. The number to the left of the decimal point in the logarithmic value is called the “characteristic” and the decimal fraction to the right of the point is called the “mantissa”. Now there’s something to impress your friends with!

This splitting of the logarithm into two parts is more than an excuse for mathematicians to introduce two jargon words. The characteristic defines the power of ten range to which the antilog belongs. So if the characteristic is 0 the range is 1 to 10, if it is 1 the range is 10 to 100 and 2 means the range 100 to 1,000. Within each range the mantissa will cycle from .0000 to .9999. So the mantissa will be exactly the same value for 5, 50, 500, etc.

Wow!

There is also another basic equation

log_{10}(A/B) = log_{10}(A) – log_{10}(B), where A/B is A divided by B

This means that we can calculate divisions by subtracting the logarithmic values, rather than adding them.

Now, there’s a thing!

One interesting outcome from the division equation above can be reached by setting the values of A and B to be the same. This means that log_{10}(A/B) equals 1 and log_{10}(A) – log_{10}(B) equals zero, since log_{10}(A) and log_{10}(B) have the same value. So the value of log_{10}(1) is zero. This result will be very helpful in a future blog – trust me!

I know what you’re thinking: “Very interesting, but I’ll stick to my calculator”. Well I’m really not suggesting that you start using log tables for multiplication and division. The reason for summarising the powers of the logarithmic function in this blog is because, as I said at the start, this function is the basis for the way that noise is measured and I will develop this theme in the next blog.

If on the other hand you are totally enthralled by the potential of this concept, you will find all that you could possibly want to know about logarithms in the Wikipedia article here.