# 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.

#### Key Points

• The natural logarithm is the logarithm with base equal to e.

• Also known as Euler's number, e is an irrational number that often appears in natural relationships in pure math and science.

• The number e and the natural logarithm have many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more.

#### Terms

• The base of the natural logarithm, 2.718281828459045…

• The logarithm in base e; either the function that given <equation contenteditable="false">$x\!$ returns $y\!$ such that $e^y = x\!$, or the value of $y\!$.

#### Figures

1. ##### Interactive Graph: Graph of the Natural Logarithm

Graph of the natural logarithm, $y=ln(x)$ or $e^y=x$. The function slowly grows to positive infinity as x increases and rapidly goes to negative infinity as x approaches 0. What shape does the graph become?

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 x increases and rapidly goes to negative infinity as x approaches 0 ("slowly" and "rapidly" as compared to any power law of x); the y-axis is an asymptote, as seen in (Figure 1).

Also known as Euler's number, e is an irrational number representing the limit of:

$(1+1/n)^n$

as n approaches infinity. In other words, e is the sum of 1 plus 1/1 plus 1/(1*2) plus 1/(1*2*3), and so on.

The number e has many applications in calculus, number theory, differential equations, complex numbers, compound interest, and more. It also is extremely useful as a base in logarithms; so useful that the logarithm with base e has its own name (natural logarithm) and symbol. Here is the proper notation for the natural logarithm of x:

$ln(x)$

The natural logarithm is so named because unlike 10, which is given value by culture and has minimal intrinsic use, e is an extremely interesting number that often "naturally" appears, especially in calculus.

The inverse of the natural log appears, for example, upon differentiating a logarithm of any base:

$\frac{d}{dx}log_b(x)=\frac{1}{x*ln(b)}$

Outside of calculus, the natural logarithm can be used to relate 1, e, i, and π, four of the most important numbers in mathematics:

$ln(-1)=iπ$

#### Key Term Glossary

asymptote
A line that a curve approaches arbitrarily closely, as they go to infinity; the limit of the curve, its tangent "at infinity".
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base
A number raised to the power of an exponent.
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complex
of a number, of the form a + bi, where a and b are real numbers and i is the square root of −1.
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complex numbers
(complex analysis) A number of the form a + bi, where a and b are real numbers and i denotes the imaginary unit
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compound
Anything made by combining several things.
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compound interest
Interest, as on a loan or a bank account, that is calculated on the total on the principal plus accumulated unpaid interest.
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constant
An identifier that is bound to an invariant value.
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e
The base of the natural logarithm, 2.718281828459045…
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equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
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exponent
The power to which a number, symbol, or expression is to be raised. For example, the 3 in $x^3$.
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function
A relation in which each element of the input is associated with exactly one element of the output.
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interest
The price paid for obtaining, or price received for providing, money or goods in a credit transaction, calculated as a fraction of the amount or value of what was borrowed.
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irrational number
Any real number that cannot be expressed as a ratio of two integers.
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logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
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natural logarithm
The logarithm in base e; either the function that given $x\!$ returns $y\!$ such that $e^y = x\!$, or the value of $y\!$.