As difficult as it may seem to believe today, there was once a dark age without calculators. Yes, people actually had to work out sums! Mathematicians, however, existed even back then, and it’s a good thing they did, or we may not be the civilisation we are today. (I’m aware that this last point is debatable, but that’s a topic for another day and another post).
Anyway, while we waited for Alan Turing to come and invent the computer, we simply HAD to know what 2100 was. And surely there were more elegant ways of doing that then multiplying 2 by itself 100 times? Fortunately, in the seventeenth century, John Napier came along, with his system of logarithms. (Or, as I remember them – thanks to Murderous Maths – log o’ rhythms.)
The basic equation for logs was, if cb = a, then logc a= b. So, if c = 10 and b = 2, a would equal 100. That means that log10 100= 2 (which is to be read as ‘log to the base 10 of 100 equals 2’). Normally, if no base is given, it’s taken to be base 10. ‘ln’ (LN in lowercase) is also used, and it means ‘natural logarithm’, or log to the base e (which equals 2.718281828459… and is the mathematical constant second in importance to π – how is it used? Again, a topic for another day).
Some rules for using logs are:
log xy = y log x
log x + log y = log (xy)
These apply to any base.
“But how can these be used to calculate 2100?” you ask.
I’m going to give you one bit of information – that log 2 ≈ 0.3 (≈ means ‘roughly equal to’). Now, log 2100 is just 100 log 2, which is equal to 100*0.3 = 30.
log 2100 = 30. By definition, this means that 2100 is about 1030. Of course, you’ll get more accurate values if you use a more accurate log value!
So, when the logs start dancing, branch out!