Logarithm

https://arbital.com/p/logarithm

by Nate Soares May 16 2016 updated Jun 20 2016


[summary: The logarithm base $~$b$~$ of a number $~$n,$~$ written $~$\log_b(n),$~$ is the answer to the question "how many times do you have to multiply 1 by $~$b$~$ to get $~$n$~$?" For example, $~$\log_{10}(1000)=3,$~$ because $~$10 \cdot 10 \cdot 10 = 1000,$~$ and $~$\log_2(16)=4$~$ because $~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$]

[summary(Technical): $~$\log_b(n)$~$ is defined to be the number $~$x$~$ such that $~$b^x = n.$~$ Thus, logarithm functions satisfy the following properties, among others:

[summary(Inverse exponentials): Logarithms are the inverse of [exponential exponentials]. That is, for any base $~$b$~$ and number $~$n,$~$ $~$\log_b(b^n) = n$~$ and $~$b^{\log_b(n)} = n.$~$]

[summary(Measure of data): A message that singles out one thing from a set of $~$n$~$ carries $~$\log(n)$~$ units of data, where the unit of information depends on the base of the logarithm. For example, a message singling out one thing from 1024 carries about three decits of data (because $~$\log_{10}(1024) \approx 3$~$), or exactly ten bits of data (because $~$\log_2(1024)=10$~$). For details, see Bit (of data).]

[summary(Generalized lengths): A quick way of approximating the logarithm base 10 is to look at the length of a number: 103 is a 3-digit number but it's almost a 2-digit number, so its logarithm (base ten) is a little higher than 2 (it's about 2.01). 981 is also a three-digit number, and it's using nearly all three of those digits, so its logarithm (base ten) is just barely lower than 3 (it's about 2.99). In this way, logarithms generalize the notion of "length," and in particular, $~$\log_b(n)$~$ measures the generalized length of the number $~$n$~$ when it's written in $~$b$~$-ary notation.]

The logarithm base $~$b$~$ of a number $~$n,$~$ written $~$\log_b(n),$~$ is the answer to the question "how many times do you have to multiply 1 by $~$b$~$ to get $~$n$~$?" For example, $~$\log_{10}(100)=2,$~$ and $~$\log_{10}(316) \approx 2.5,$~$ because $~$316 \approx$~$ $~$10 \cdot 10 \cdot \sqrt{10},$~$ and [ multiplying by $~$\sqrt{10}$~$ corresponds to multiplying by 10 "half a time"].

In other words, $~$\log_b(x)$~$ counts the number of $~$b$~$-factors in $~$x$~$. For example, $~$\log_2(100)$~$ counts the number of "doublings" in the number 100, and $~$6 < \log_2(100) < 7$~$ because scaling an object up by a factor of 100 requires more than 6 (but less than 7) doublings. For an introduction to logarithms, see the Arbital logarithm tutorial. For an advanced introduction, see the [advanced_log_tutorial advanced logarithm tutorial].

Formally, $~$\log_b(n)$~$ is defined to be the number $~$x$~$ such that $~$b^x = n,$~$ where $~$b$~$ and $~$n$~$ are numbers. $~$b$~$ is called the "base" of the logarithm, and has a relationship to the [number_base base of a number system]. For a discussion of common and useful bases for logarithms, see the page on [logarithm_bases logarithm bases]. $~$x$~$ is unique if by "number" we mean [4bc $~$\mathbb R$~$], but may not be unique if by "number" we mean [complex_number $~$\mathbb C$~$]. For details, see the page on [complex_logarithm complex logarithms].

Basic properties

Logarithms satisfy a number of desirable properties, including:

For an expanded list of properties, explanations of what they mean, and the reasons for why they hold, see Logarithmic identities.

Interpretations

Applications

Logarithms are ubiquitous in many fields, including mathematics, physics, computer science, cognitive science, and artificial intelligence, to name a few. For example:


Comments

Eric Bruylant

The logarithm base $~$b$~$ of a number $~$n,$~$ written $~$\\log\_b(n),$~$ is the answer to the question "how many times do you have to multiply 1 by $~$b$~$ to get $~$n$~$?" For example, $~$\\log\_{10}(100)\=2,$~$ and $~$\\log\_{10}(316) \\approx 2.5,$~$ because $~$316 \\approx$~$ $~$10 \\cdot 10 \\cdot \\sqrt{10},$~$ and multiplying by $~$\\sqrt{10}$~$ corresponds to multiplying by 10 "half a time"\.

Having a long redlink which does not point anywhere seems weird? Does the page it should point to now exist?

Nate Soares

No (and it won't, until someone starts writing good explanations of radicals). I think it's fine to have redlinks to nowhere, when it's not clear yet which page will explain the concept.