e

(noun)

The base of the natural logarithm, approximately 2.718281828459045…

Related Terms

  • logarithm
  • natural logarithm

(noun)

The base of the natural logarithm, 2.718281828459045…

Related Terms

  • logarithm
  • natural logarithm

Examples of e in the following topics:

  • Graphs of Exponential Functions, Base e

    • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
    • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$ where $e$ is the number (approximately 2.718281828) described previously.
    • The graph's $y$-intercept is the point $(0,1)$, and it also contains the point $(1,e).$ Sometimes it is written as $y=\exp (x)$.
    • The graph of $y=e^{x}$ is upward-sloping, and increases faster as $x$ increases.
    • $y=e^x$ is the only function with this property.
  • The Number e

    • Note that $\ln (e) =1$ and that $\ln (1)=0$.
    • There are a number of different definitions of the number $e$.
    • Another is that $e$ is the unique number so that the area under the curve $y=1/x$ from $x=1$ to $x=e$ is $1$ square unit.
    • Another definition of $e$ involves the infinite series $1+\frac{1}{1!}
    • It can be shown that the sum of this series is $e$.
  • Conics in Polar Coordinates

    • With this definition, we may now define a conic in terms of the directrix: $x=±p$, the eccentricity $e$, and the angle $\theta$.
    • For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
    • For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:
  • Cramer's Rule

  • Eccentricity

    • The eccentricity, denoted $e$, is a parameter associated with every conic section.
    • The value of $e$ is constant for any conic section.
    • The value of $e$ can be used to determine the type of conic section as well:
  • 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.
    • The natural logarithm is the logarithm with base equal to e.
    • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$, also arises in naturally in many contexts.
    • Its value at $x=1$ is $0$, while its value at $x=e$ is $1$.
  • Applications of Hyperbolas

    • However, this is the very special case when the total energy $E$ is exactly the minimum escape energy, so $E$ in this case is considered to be zero.
    • Blue is a hyperbolic trajectory ($e > 1$).
    • Green is a parabolic trajectory ($e = 1$).
    • Red is an elliptical orbit ($e < 1$).
    • Grey is a circular orbit ($e = 0$).
  • Scientific Notation

    • Because superscripted exponents like $10^7$ cannot always be conveniently displayed, the letter E or e is often used to represent the phrase "times ten raised to the power of" (which would be written as "$\cdot 10^n$") and is followed by the value of the exponent.
    • Note that in this usage, the character e is not related to the mathematical constant $\mathbf{e}$ or the exponential function $e^x$ (a confusion that is less likely if a capital E is used), and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation.
  • Introduction to Exponential and Logarithmic Functions

    • Let us consider instead the natural log (a logarithm of the base $e$). 
    • The natural logarithm is the inverse of the exponential function $f(x)=e^x$.
    • It is defined for $e>0$, and satisfies $f^{-1}(x)=lnx $.
    • That is, $e^{lnx}=lne^x=x$.
  • Parts of an Ellipse

    • All conic sections have an eccentricity value, denoted $e$.
    • All ellipses have eccentricities in the range $0 \leq e < 1$.
    • $\displaystyle{ \begin{aligned} e &= \frac{\sqrt{a^2 - b^2}}{a} \\ &= \sqrt{ \frac{a^2 - b^2}{a^2} } \\ &= \sqrt{ 1 - \frac{b^2}{a^2}} \end{aligned} }$
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