What is the number e and how is it related to logarithms?

In summary, the number "e" can be defined as the value for which the derivative of e^x is equal to e^x. This is also one of the main reasons for its frequent use. When differentiating logarithms, using e as the base simplifies the process as the derivative is just 1/(x*ln(e)). Understanding the origins and purpose of e is important, after which memorizing the derivatives becomes easier.
  • #1
J7
10
0
The Number "e" and Logarithms

Hi, I'm having a lot of difficulty understanding the number "e" and logarithms, especially in terms of differentiating them. Is it just a matter of memorization or are there tricks to finding the derivatives? Help!
 
Physics news on Phys.org
  • #2
Yes,u better memorize them.It's much better than deducing it with every occasion (:yuck:)...

Thery're the simplest diff.rules possible.

Daniel.
 
  • #3
There are different ways of defining and introducing e, but when you're new to logarithms, I think the easiest way is to define e as that number for which:
[tex]\frac{d}{dx}e^x=e^x[/tex].

More detail:

[tex]\frac{d}{dx}a^x=\lim_{h \to 0} \frac{a^{x+h}-a^x}{h}=a^x\lim_{h \to 0}\frac{a^h-1}{h}[/tex]

You can check the limit exists (for a>0 ofcourse). Also, if [itex]f(x)=a^x[/itex], then you can see that [itex]f'(x)=a^xf'(0)[/itex]
If you try some values for a and some small values for h to get some sight as to the value of the limit :

If a=2, then the limit is approx. 0.6934
If a=3, then the limit is approx. 1.0986

The larger a, the larger the limit. There exists some number between 2 and 3 for which is value is 1. You can define that number to be e.
So e is that number for which:
[tex]\lim_{h \to 0}\frac{e^h-1}{h}=1[/tex]

Therefore you have this nice rule when differentiating e^x:
[tex]\frac{d}{dx}e^x=e^x[/tex]
and this is actually one of the main reasons it is used so often.

Likewise, differentiating [itex]\log_a x[/itex] gives [itex]\frac{1}{x \ln a}[/itex], so if you use base e (the natural logarithm) the derivative is simplified.

The above hopefully gives some insight into e. It's important to understand where it comes from and why it is used so much. After that, memorizing the derivatives and such is trivial.
 
Last edited:

1. What is the number e?

The number e is a mathematical constant that is approximately equal to 2.71828. It is also known as Euler's number or Napier's constant. It is an important number in calculus and other areas of mathematics.

2. How is the number e calculated?

The number e is calculated using the following infinite series: e = 1 + 1/1! + 1/2! + 1/3! + ...

3. What are logarithms?

Logarithms are mathematical functions that are used to solve exponential equations. They are the inverse of exponential functions and are denoted by the symbol log. For example, log base 10 of 100 is equal to 2, because 10 raised to the power of 2 is equal to 100.

4. What is the relationship between e and logarithms?

The natural logarithm, denoted by ln, is the inverse of the exponential function with base e. This means that ln(e) = 1 and e^ln(x) = x. Logarithms with other bases can be converted to natural logarithms by using the following formula: log base a of x = ln(x)/ln(a).

5. How are e and logarithms used in real life?

The number e and logarithms have many applications in science, engineering, and finance. They are used to model exponential growth and decay, calculate compound interest, and solve differential equations. They are also used in data analysis and signal processing.

Similar threads

  • Calculus
Replies
3
Views
1K
Replies
5
Views
1K
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
10
Views
1K
Replies
4
Views
696
Replies
2
Views
769
  • Differential Equations
Replies
2
Views
2K
  • Calculus
Replies
4
Views
1K
Replies
2
Views
4K
Back
Top