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Mathematics - Data Science - Computer Science

Overview
Calculus
Computing Systems
Data
Graphs
Linear Algebra
Models and Modeling
Organization
Probability
Programming Constructs
- Programming Constructs Overview
- Abstraction
- Array
- Attribute
- Best-first Search
- Binary Search
- Block
- Branch
- Callback
- Class
- Conditional
- Constructor
- Container/Collection
- Dynamic Array
- Encapsulation
- Exception
- Expression
- Function
- Garbage Collection
- Greedy Algorithms
- Hash
- HTTP Request
- Identifier
- Inheritance
- Inner Class
- Instance
- Iterator
- Keyword
- Lambda
- Libraries
- List
- Linked List
- Literal
- Metaclass
- Method
- Mixin
- Object
- Operator
- Overloading
- Overriding
- Package
- Parameter
- Polymorphism
- Primitive
- Programming Process
- Recursion
- Reflection
- Regular Expression
- Reserved Word
- Return
- Sort
- Statement
- Switch
- Table
- This/Self
- Token
- Type
- Variable
Statistics
Trigonometry

Glossary and Index
Mathematical Symbols
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Mathematical Notation Powered by CodeCogs

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Logarithms

A logarithm is an exponent with a given base. Consider the equations below:

$\\ if \,\,\,\, x=b^{y}\,\,\,\, then \,\,\,\, log_{b}\left ( x \right )=y \,\,\,\,\,\,\,\,\, \textup{where:} \\ \\ y \,\,\,\,\,\textup{the exponent/logarithm - a real number } \\ b \,\,\,\textup{\,\,\,the base - a positive real number not equal to 1, typically 2, 10, or} \,e \\ x \,\,\textup{\,\,\,a positive real number} \\ \\ \textup{An example is:} \\ \\ if \,\,\,\,\, 64 = 2^{6}\,\,\,\,then\,\,\,\,log_{2}\left (64 \right )=6$

Key aspects of logarithms include:

a logarithm is the inverse function to exponentiation
in the simplest case, the logarithm counts the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3
logarithms can be used to compress a scale of measurement

Rules

Following are mathematical rules for working with logarithms:

$\\ log_{b}\left ( b^{x} \right )=x \\\\log_{b}\left ( x\ast y \right )=\log_{b}\left ( x \right )+\log_{b}\left ( y \right ) \\\\log_{b}\left ( x/ y \right )=\log_{b}\left ( x \right )-\log_{b}\left ( y \right ) \\\\log_{b}\left ( x^{y} \right )=y\ast log_{b}\left ( x \right ) \\\\log_{b}\left ( x \right )=\frac{log_{c}\left ( x \right )}{log_{c}\left ( b \right )}$

An Example

An example of the use of logarithms is displaying exponential growth, such as Moore’s Law:

References

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