指数函数和$e$

(Video) Intro to Base $e$ Exponential Function

by Carolee Pederson

Carolee Pederson has a good video introduction to exponential functions.

Now that we've learned about Benford's Law, let's move onto the next lesson in log applications.

Euler's Number

Euler's number, $e$, is a mathematical constant we commonly use with exponents and logarithms. It's one of the most important constants in mathematics. Like $\pi$, it is an irrational number, meaning that its decimal expansion goes on forever without repeating.

It may be helpful to memorize that $e=2.71828$.

We use $e$ a lot in natural logs.

The number $e$ occurs naturally in situations involving continuous growth and decay, making it fundamental to algebra, calculus, finances, and so many other areas!

Euler's number is the base of the natural exponential function: $$f(x) = e^x$$

This function has many unique and useful properties. The number is named after Leonhard Euler, the mathematician who helped popularize its use, though he didn't originally discover it.

Some important values to memorize:

$$e^0=1$$ $$e^1=e$$ $$e^2=7.389$$ $$e^{-1}=0.368$$

Now let's look at the graph of $y=e^x$.

The graph of $y=e^x$:

• Passes through the point $(0,1)$
• Increases rapidly (exponentially!) as $x$ increases
• Has a horizontal asymptote at $y=0$

Looking Further

You can also use Euler's Formula on complex numbers.

The inverse of the exponential function $e^x$ is the natural logarithm.

Euler's Number

Euler's number, written as $e$, is one of the most important constants in math. It is a mathematical constant with an approximate value of:

$$e = 2.718281828...$$

The digits continue forever beyond the decimal point without repeating because the number is an irrational number. Like $\pi$, its exact decimal expansion never ends.

The constant is often called Euler's constant and serves as the base of the natural logarithm, written as $\ln$. It comes from historical work on compound interest. In the late 1600s, Jacob Bernoulli discovered that increasing the number of compound intervals, while keeping the same interest rate, caused an investment to approach a fixed limit. As the number of compounding periods per years gets larger, the value approaches $e$.

Later, Leonhard Euler studied this constant and introduced the notation $e$, which is why it now bears his name.

####Continuous Growth

One of the main reasons $e$ is so important is because it models exponential growth that occurs continuously. For continuously compounded interest, the amount in an account is given by:

$$A = Pe^{rt}$$

, where $P$ is the initial principal, $r$ is the annual interest rate, and $t$ is the time in years. This formula describes money that is growing exponentially because interest is added continuously rather than at fixed intervals.

####Other Ways to Define $e$

The compound interest definition is not the only way to describe $e$. It can also be defined using an infinite series. In calculus, $e$ also appears naturally in problems involving the area under the curve of the function $y=\frac{1}{x}$.

####The Function

The exponential function $y=e^x$ has several properties. It models natural growth processes and has a graph that:

• Passes through the point $(0,1)$
• Increases continuously
• Approaches $0$ as $x$ approaches negative infinity

Because of these properties, $e$ appears throughout math, science, economics, and statistics.