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Natural Logarithms
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
Learning Objective

Identify some properties and uses of the natural logarithm
Key Points
Terms

e
The base of the natural logarithm, 2.718281828459045…

natural logarithm
The logarithm in base e; either the function that given
$x\!$ returns$y\!$ such that$e^y = x\!$ , or the value of$y\!$ .
Full Text
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. The natural logarithm is the logarithm with base equal to e.
The function slowly grows to positive infinity as
The graphs of $\log_2 x$ , $\ln x$ , and $\log_3 x$ .
The graph of the natural logarithm lies between the base 2 and the base 3 logarithms.
The natural logarithm is sometimes written as
Just as the exponential function with base
The natural logarithm function can be used to solve equations in which the variable is in an exponent. For example, one way to find the positive root of the equation
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Key Term Reference
 Base
 Appears in these related concepts: Common Bases of Logarithms, Negative Exponents, and Simplifying Exponential Expressions
 Interest
 Appears in these related concepts: Accounting for Interest Earned and Principal at Maturity, Interest Compounded Continuously, and Tax Considerations
 asymptote
 Appears in these related concepts: Graphs of Exponential Functions, Base e, What Are Conic Sections?, and Types of Conic Sections
 base
 Appears in these related concepts: Changing Logarithmic Bases, Logarithms of Powers, and Rational Exponents
 complex
 Appears in these related concepts: Introduction to Complex Numbers, Electron Transport Chain, and Transduction of Sound
 complex numbers
 Appears in these related concepts: Addition, Subtraction, and Multiplication, Phasors, and Imaginary Numbers
 compound interest
 Appears in these related concepts: Calculating Future Value, Calculating Present Value, and MultiPeriod Investment
 constant
 Appears in these related concepts: Inverse Variation, Graphing Quadratic Equations in Vertex Form, and Direct Variation
 doubling time
 Appears in these related concepts: ProblemSolving, The Energetics of Chemolithotrophy, and Generation Time
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Graphs of Equations as Graphs of Solutions
 exponent
 Appears in these related concepts: Logarithms of Products, Logarithms of Quotients, and Rules for Exponent Arithmetic
 exponential
 Appears in these related concepts: Exponential Growth and Decay, Solving General Problems with Logarithms and Exponents, and Population Growth
 exponential function
 Appears in these related concepts: The Natural Exponential Function: Differentiation and Integration, Further Transcendental Functions, and Basics of Graphing Exponential Functions
 function
 Appears in these related concepts: Solving Differential Equations, Visualizing Domain and Range, and The Vertical Line Test
 graph
 Appears in these related concepts: Reading Points on a Graph, Graphing Equations, and Graphical Representations of Functions
 halflife
 Appears in these related concepts: Exponential Decay, HalfLife of Radioactive Decay, and Isotopes
 inverse function
 Appears in these related concepts: Introduction to Inverse Functions, Introduction to Exponential and Logarithmic Functions, and Inverse Trigonometric Functions
 logarithm
 Appears in these related concepts: Solving Problems with Logarithmic Graphs, Graphs of Logarithmic Functions, and The Number e
 logarithmic function
 Appears in these related concepts: Restricting Domains to Find Inverses and Limited Growth
 root
 Appears in these related concepts: Newton's Method, Radical Equations, and Introduction to Radicals
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 variable
 Appears in these related concepts: What is a Linear Function?, Math Review, and Introduction to Variables
Sources
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Cite This Source
Source: Boundless. “Natural Logarithms.” Boundless Algebra. Boundless, 06 Jul. 2016. Retrieved 28 Jul. 2016 from https://www.boundless.com/algebra/textbooks/boundlessalgebratextbook/exponentslogarithmsandinversefunctions8/therealnumbere356/naturallogarithms1775881/