Logarithms | Additive functions | Elementary special functions
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation. The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science. Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because the logarithm of a product is the sum of the logarithms of the factors: provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However, in general settings, the logarithm tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete logarithm is the multi-valued inverse of the exponential function in finite groups; it has uses in public-key cryptography. (Wikipedia).
Ex: Determine the Value of a Number on a Logarithmic Scale (Log Form)
This video explains how to determine the value of several numbers on a logarithmic scale scaled in logarithmic form. http://mathispower4u.com
From playlist Using the Definition of a Logarithm
Solving the Logarithmic Equation log(A) = log(B) - C*log(x) for A
Solving the Logarithmic Equation log(A) = log(B) - C*log(x) for A Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys
From playlist Logarithmic Equations
What are natural logarithms and their properties
👉 Learn all about the properties of logarithms. The logarithm of a number say a to the base of another number say b is a number say n which when raised as a power of b gives a. (i.e. log [base b] (a) = n means that b^n = a). The logarithm of a negative number is not defined. (i.e. it is no
From playlist Rules of Logarithms
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From playlist Exponential and Logarithmic Expressions and Equations
What is a Logarithm : Logarithms, Lesson 1
This tutorial explains a practical way to think about logarithms. Join this channel to get access to perks: https://www.youtube.com/channel/UCn2SbZWi4yTkmPUj5wnbfoA/join :)
From playlist All About Logarithms
What are the properties of logarithms and natural logarithms
👉 Learn all about the properties of logarithms. The logarithm of a number say a to the base of another number say b is a number say n which when raised as a power of b gives a. (i.e. log [base b] (a) = n means that b^n = a). The logarithm of a negative number is not defined. (i.e. it is no
From playlist Rules of Logarithms
Properties of Logarithms : Logarithms, Lesson 5
This tutorial shows how a logarithm containing a product in its argument can be written as a sum of two logarithms, and how a logarithms of a quotient can be written as a subtraction of two logarithms. Join this channel to get access to perks: https://www.youtube.com/channel/UCn2SbZWi4yTk
From playlist All About Logarithms
Solving a logarithim, log81 (x) = 3/4
👉 Learn how to solve logarithmic equations. Logarithmic equations are equations with logarithms in them. To solve a logarithmic equation, we first isolate the logarithm part of the equation. After we have isolated the logarithm part of the equation, we then get rid of the logarithm. This i
From playlist Solve Logarithmic Equations
Solving a natural logarithmic equation using your calculator
👉 Learn how to solve logarithmic equations. Logarithmic equations are equations with logarithms in them. To solve a logarithmic equation, we first isolate the logarithm part of the equation. After we have isolated the logarithm part of the equation, we then get rid of the logarithm. This i
From playlist Solve Logarithmic Equations
Introduction to Solving Logarithms and Exponentials (Precalculus - College Algebra 57)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com How to solve logarithms by using exponentials (without common bases) and how to solve exponentials by using logarithms (without common bases). The focus of the video will be o
From playlist Precalculus - College Algebra/Trigonometry
Introduction to Logarithms and Their Graphs (Precalculus - College Algebra 55)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com A study of logarithms from the vantage point of being the inverse of an exponential. Focus will be on the creation of the graph of the logarithm and how to change between log
From playlist Precalculus - College Algebra/Trigonometry
Solving Logarithms with Common Bases (Precalculus - College Algebra 62)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com How to solve logarithmic equations with no constants and only logarithms by setting the arguments of equaled based logarithms equal.
From playlist Precalculus - College Algebra/Trigonometry
Defining the Natural Logarithm as an Integral?!?!?
We typically define ln(x), the natural logarithm of x, by first defining the exponential function of x, noting that this is a 1:1 function, and then defining ln(x) as the inverse function to exponential. In this video we go the other way around. We define ln(x) as a particular integral, an
From playlist Calculus II (Integration Methods, Series, Parametric/Polar, Vectors) **Full Course**
Expanding logarithmic expressions
👉 Learn how to expand logarithms using the product/quotient rule. The product rule of logarithms states that the logarithm of a product to a given base is equivalent to the sum of the logarithms of the terms that make up the product to the same base. For example: log ab = log a + log b. Th
From playlist Power to Quotient Rule of Logarithms
Expanding logarithmic expressions
👉 Learn how to expand logarithms using the product/quotient rule. The product rule of logarithms states that the logarithm of a product to a given base is equivalent to the sum of the logarithms of the terms that make up the product to the same base. For example: log ab = log a + log b. Th
From playlist Power to Quotient Rule of Logarithms
How to Expand Logarithms (Precalculus - College Algebra 59)
Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com How to use the properties of logarithms to expand logarithmic expressions.
From playlist Precalculus - College Algebra/Trigonometry
How to expand logarithmic expressions to multiple logarithms
👉 Learn how to expand logarithms using the product/quotient rule. The product rule of logarithms states that the logarithm of a product to a given base is equivalent to the sum of the logarithms of the terms that make up the product to the same base. For example: log ab = log a + log b. Th
From playlist Power to Quotient Rule of Logarithms
How to expand a log expression using the rules of logarithms
👉 Learn how to expand logarithms using the product/quotient rule. The product rule of logarithms states that the logarithm of a product to a given base is equivalent to the sum of the logarithms of the terms that make up the product to the same base. For example: log ab = log a + log b. Th
From playlist Power to Quotient Rule of Logarithms
Learn the basics to expanding a logarithmic expression
👉 Learn how to expand logarithms using the product/quotient rule. The product rule of logarithms states that the logarithm of a product to a given base is equivalent to the sum of the logarithms of the terms that make up the product to the same base. For example: log ab = log a + log b. Th
From playlist Power to Quotient Rule of Logarithms
What are the properties of logarithms
👉 Learn all about the properties of logarithms. The logarithm of a number say a to the base of another number say b is a number say n which when raised as a power of b gives a. (i.e. log [base b] (a) = n means that b^n = a). The logarithm of a negative number is not defined. (i.e. it is no
From playlist Rules of Logarithms