What is it about the number e that makes it so that taking log to the base e is called the natural logarithm?
In: Mathematics
The natural log tells you how long 100% **continuous*** growth takes. If you have like, cells that divide (everything doubles), that’s 100% growth. That’s what’s natural about this: it comes up a lot in things that grow.
Say you have a really good savings account that doubles every year, 100% continuous growth. You start with $1 in it. How long until you have $10? `ln(10) = 2.3 years`. And if I have one of these accounts, and I want to know how much my $3 will grow in 5 years, well that’s `3e^5 = $445`.
EDIT: * – see the convo… I should have clarified how I’m calculating interest. Math good, words bad.
I believe it is because log base e and e as an exponent have nice derivatives/integrals and are easy to work with in calculus applications.
You could y=e^x the “natural exponent” instead of “exponent of x base e” which is correct. But we have a faster way of saying this “e to the x”
x=ln(y) is the reverse of that function and is called the “natural logarithm” instead of “logarithm of y base e”.
Humans like to say things quickly when we have to say things a lot.
We use base-10 logs sometimes because we have 10 fingers and count in tens. log base-10 tells us the magnitude of a number in powers of how many fingers we have. Any other log base would be equally arbitrary. What would be a “natural” base, one free of human biases?
Well, mathematicians decided “e” would do that nicely. e can be calculated in lots of mathematical ways that don’t depend on ten fingered creatures and decimal arithmetic, for example as the infinite sum of inverse factorials. “e” appears “naturally” in a lot of mathematical areas, which we’d expect seven-fingered aliens who count in base-14 to also find. For example the rate of change of e^x (e to the power of x) is e^x, and the rate of change of the natural log function is 1/x, which looks simple and a “natural” choice for log base. Other bases end up with constants and scale factors when you try and compute the rates of change (slope, gradient, differential).
it is just a name, just like imaginary numbers are not imaginary, and dark matter and dark energy might not be the best description either.
Short answer is that’s just a name. There’s nothing special mathematically about the name aside from a feeling.
Long answer is, there are often things in math we call canonical or natural. These are somehow special in some fundamental way. e is one such number. Consider the following functional form:
y = a^x
If you take the derivative of this, you get y’ = ln(a) a^x . Seems a bit complicated doesn’t it? But if your base were e instead of a, then you’d get y is its own derivative. In fact e shows up in seemingly different and unrelated fields in a similar way. This makes mathematicians feel there’s something special about e.
Given that ln is e’s inverse, it therefore gets the honorific of “natural” log.
e^x is a special equation. For every value X, its value, its slope (how fast it’s going up), and its integral (area under the curve, or aggregate effect over time) are all e^x. This makes natural exponents and algorithms a lovely equation to pair with calculus, which is used commonly in physics, finance, and generally studying the effect of things over time.
I’m not sure why it’s called natural, but the reason e is special is because with exponetiation it is equal to its own derivative.
So for f(x)= e^x
then f'(x) = e^x
with doesn’t seem that useful
But since for any number (by simple exponetiation rules)
a = e^ln(a)
but then for f(x) = a^x = e^ln(a)x
and then by a simple chain rule then
f'(x) = ln(a)e^ln(a)x = ln(a)a^x
making e a very useful constant.
First: A logarithm must have a base associated with it. Let’s say that base is 10. Then taking the base 10 logarithm of a number is the answer to the question, To what power should I raise 10 to get this number? So if you are taking the base 10 logarithm of 100, the answer is 2, because 10 raised to the power of 2 equals 100.
The number e shows up in a lot of different contexts, but one of the simplest ones is that the function e^x has a rate of change that is its value. This makes it a simple to use base for describing all kinds of things where the amount they are changing is based on their value, such as population growth, growth due to interest rates, air resistance and more.
So the function e^x shows up so much we give it it’s own name, the exponential function. And because it shows up so much we are often trying to solve for x where e^x = some number. So we need to take the logarithm of both sides of our equation to answer this, and the most natural base to use for this is the number that is being raised to the power of x, which is e. So we use a logarithm with a base of e and we call it the natural logarithm. It also gets called “log base e”, but since it shows up so often people just call it the natural log.
Often, complicated mathematical equations and operations can “cancel out” if you use e as the log base.
Latest Answers