What is a natural logarithm? Why is ln(1) = 0?
In: Mathematics
There are some great other answers explaining why, based on defining logs as the inverse of exponentials, ln(1) must be 0. But here’s a good example of why we get into trouble if we don’t.
Logarithms are a maths thing, with a bunch of rules around them. In particular we have the addition rule:
> ln(a.b) = ln(a) + ln(b)
This is the log version of our basic power rule:
> x^(a+b) = x^(a).x^(b)
and until recently it was the main point of dealing with logs; logs turn multiplication (which can be difficult) into addition (which is easier).
Before electronic calculators people used to use “log tables” to do hard multiplication. Say you wanted to do 4,454,230 x 6,745,234; you would look up log(4,454,230) and log(6,745,234) in your table, add those numbers together, and then look up that number on the other side of the log table, and it would give you the answer you want.
But if ln(1) isn’t 0 we get into trouble:
> ln(1) = ln(1 x 1) = ln(1) + ln (1) = 2ln(1)
Essentially we can always add ln(1) onto a log without changing the number. If it isn’t 0 that is going to be a problem.
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