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Learn Logarithms in 5 Minutes

Learn Logarithms in 5 Minutes

Suppose you have a number x and you decide to represent it as 2 to some power. That is, you find a number y such that..


Mathematicians say that "y is the logarithm of x" and write..


What about the number 2? It's called the "base" of the logarithm, so technically we should really say..

"y is the logarithm of x base 2"

Obviously you can pick any base you like, and the two most popular are 10 and e, where e is a famous number we discuss in another post. Logarithms base 10 are called "common logarithms" and usually written as log(x). Logarithms base e are called "natural logarithms" and usually written as ln(x). But that's just a convention, technically speaking you should always specify which base you're using.

Let's try an example, what's log(1)? In other words, what power do I have to raise 10 by to get the result of 1? The answer is zero, so..

log(1)=0 because 10^0=1

What's log(0)? It's not defined, because it's impossible to raise 10 to a power and get the result zero! This is a good reminder, a function does not have to be defined for all values of x. At x=0 log(x) is simply not defined.

Here's another example which illustrates an interesting point..

log(1000)=3 because 10^3=1000

This shows a famous feature of the logarithm function, as x increases the logarithm of x increases very, very slowly. In fact, the logarithm is an amazingly slow function.

Another famous feature of the logarithm function is..


which is true for any base.

Logarithms show up in many areas on math, especially in number theory.

For example, the simple sum..


is approximately equal to log(n), and the approximation gets better and better as n increases.

In physics logarithms show up in the calculation of entropy and are also very important in information theory. For some reason the logarithm function seems to be associated with measurement of randomness and information content of a system.

Content written and posted by Ken Abbott