What is a natural logarithm? Why is ln(1) = 0?
In: Mathematics
Might i suggest r/learnmath for better help?
log(1) = 0 is true of all logs, because log is the inverse of exponent, and any number to the power of zero is 1.
It can be a little confusing, but “the log in base X of Y equals Z” is the same as saying “X to the power of Z equals Y.” You may have to read that several times and maybe even write it out for it to sink in.
The natural log, usually written ln (but to many programmers and mathematicians “log” by default means ln), is the log using base e. “e” is a special number, sort of like pi, that is special because of how it relates to rates of change. It’s about equal to 2.718 plus infinite digits after that.
The function e^x is special because if you graph it, the rate of change at every point is also equal to e^x. The function describes its own rate of change.
This is useful when modeling situations where the rate of change depends on how much of something you currently have.
For example, gaining interest on money. If you have more money, you earn more interest so your amount of money increases faster, and then you have more money and get even more interest and get even more money…
How fast the money grows depends on how much money there currently is. The rate of change depends on the amount, and the number e is special because when you use it as a base, the rate of change is *exactly equal* to the current amount.
It’s the inverse of exponentiation, specifically with base e (that’s why it’s “natural” — can expand more on that if you wish). i.e.
y = e^x <–> ln(y) = x
Plug in x=0, y=1 and you’ll see that ln(1) = 0 is essentially the same statement as e^0 = 1. In fact, the log of 1 with any base is equal to 0 for the same reason (any number raised to the power 0 is 1).
When you take a logarithm you’re asking “what power do I need to raise this base to, in order to get the given answer?”. For example, log2(8)=3, because 2³ =8.
You can take logs in any base, but the natural log uses the mathematical constant *e* as its base (~2.718).
As for ln(1)=0, this actually works in any base, as anything to the power of zero is equal to 1.
Lets use another equation:
e^0=1
Well you might know that already as everynumber to the power of 0 =1. we pretend we don’t.
e = eulers number is a special number like Pi for example. it has an opposite just like multiplication is the opposite of divison.
This opposite is ln. the logarithm naturalis.
using it on the equation gets us
ln(e^0)=ln(1)
ln and e cancel eachother out and the zero moves down. so we get:
ln(1)=0
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.
ln(x) is asking the question e to what power is x
So ln(1) is asking e to what power is 1?
Well e^0 = 1, so ln(1)=0
2 to the power of what makes 16?
2⁴ = 16 so the solution to that question is 4. We write that as log(16) = 4.
3 to the power of what makes 16?
Its a bit more complicated but we also call the number that solves this question log(16). However, the fact that we are taking powers of 3 this time obviously makes this number different to the other “log(16)”. So to resolve that issue, we call this one “log to the base of 3” and the other one “log to the base of 2”. Usually, you just write “log(16)” and add the base as a small subscript.
Now, “log(1) to the base of 2” simply means
2 to the power of what makes 1?
Since everything to the power of 0 is one, we actually know that log(1) = 0 *regardless of base* .
Next, the natural logarithm is just the logarithm where the base is a special number called “e” or eulers constant. Its about 2.71… and due to some calculus reasons, its a very handy base (and in some sense the “natural” choice of base). Still, ln(1) = 0 remains true.
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