What are the purposes of “e” and ln(x) in calculus?

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i know a logarithm is the inverse of exponents… but i just cannot fathom how these are supposed to work in calculus, why we’re deriving them, what “e” means, why it’s important etc.

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for reference currently taking calculus ab as well

In: Mathematics
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e is a special constant like pi that has important properties in certain contexts.

What’s really special about e^x is that it is its own derivative. When you take the derivative of something like x^5 , you end up with 5x^4 . When you take the derivative of e^x you get e^x . It’s the function f(x) where f(x) = f'(x). That means that if you ever need to do calculus with exponentials, you will generally have an easier time if you express them as e^something .

they’re very important because they make derivatives, integrals and differential equations easy to manipulate.

Their use in calculus is mostly about defining differential equations solutions and calculating crazy intergrals.

In physics they’re everywhere, since they’re used to find solutions to differential equations and physics is all about differential equations.

And the answer to why exp(x) is so important, it’s not important, it’s super convenient because its properties (equal to its derivative) simplify math enough to find easy solutions to complicated problems.

e is the value that ties in with growth where the rate of growth at a given is proportional to the current size and occurs continually, rather than incrementally.

This is represented by e^x

The rate of growth/slope of the curve is equal to the value of the curve at any location, and the area under the curve between 0 and any location is equal to the value of the curve at that location.

Ie: the growth is equal to the population

This can be manipulated with constants ie e^kx for other growth rates, but in general anything with continuous growth is going to have an equation involving e

ln is the log in base e

I feel the historical context behind it is kind of important.

When the smart people in the 1600s invented calculus, they started trying to find derivatives for various functions. They found derivatives of many common functions but then they started messing around with exponential functions, functions like 2^x and 3^x .

They would graph it by hand and try to manually see if they can get the derivative through approximation, and usually what they found was that the derivative was proportional to the original function of 2^x and 3^x. In other words, the derivative for these functions ended up being something like 2^x * C for some weird constant C that no one understood. They also used the direct definition of the derivative which showed something similar, the result was 2^x * C for some weird limit (lim h approaches 0 of (2^h – 1)/h, which we now know is ln 2).

This brought up two questions.

1. What is C?

2. Is there a value of a for a^x where C = 1? In other words, is there a function where the derivative is equal to itself.

Now these questions were actually in some sense the same, if we can find this magical value of a for which C = 1, then we can use the exponent change of base formula to put all functions in the form of a^x into the proper exponent.

Either way, this number turned out to be a curious constant known as e. The first hint was that C happened to be less than 1 for a = 2 and more than 1 for a = 3, so a where C = 0 is probably between those two. Eventually we were able to derive a formula to describe e more appropriately.

And C ended up being ln a. Logarithm base e of a.