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
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.
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