Exponentials and Logarithms¶
It turns out that there's a special base, the number
The Basics¶
Before you start doing calculus with exponentials, you really need to understand their properties.
Exponentials¶
The rough idea is that we'll take a positive number, called the base
, and raise it to a power called the exponent
:
For any base
Logarithms¶
Here are the rules valid for any base
Definition of ¶
The
A question about compound interest¶
A long time ago, a dude named Bernoulli answers a question about compound interest. Here is the setup for his question. Let's suppose you have a bank account that pays interest at a generous rate of
Now suppose you find another bank that also offers an annual interest rate of
The second account is a little better than the first. It makes sense when you think about it--compounding is beneficial, so compounding more often at the same annual rate should be better. Let's try 3 times a year at the annual rate of
The answer to our question¶
Let's turn to some symbols. First, let's suppose that we are compounding n times a year at an annual rate of
We would like to know what happens in the limit as
If this limit turns out to be finite, then by compounding more and more often, you could get more and more money in a single year. On the other hand, if it turns out to be finite, we'll have to conclude that there is a limitation on how much we can increase our fortune with an annual interest rate of
First, lets' set
Use the exponential rule:
Let's pull a huge rabbit out of the hat and set:
and we have:
The number
The base natural logarithm
. Since we have a new base
Differentiation of Logs and Exponentials¶
With the definition of limit, we can find the derivatives of logs and exponentials:
Behavior of exponentials near 0¶
In fact, we know that:
But what about the limit:
Well, try setting
Simplify the equation, we have:
Behavior of logarithms near 1¶
It turns out that the situation is pretty simillar to the case of exponentials near 0:
Behavior of exponentials near or ¶
Let's take a look at the graph of
and get the conclusion:
and
What if we the base
In addtion to all equations above, as Exponentials grow quickly
:
no matter how large
Behavior of logs near ¶
The sage continues. Here is the graph of
And as the logs grow slowly, if
no matter how small
Behavior of logs near 0¶
Logs grow slowly at
no matter how small