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Exponential

The exponential function is a fundamental mathematical function of the form f(x) = a^x, where a > 0, a \neq 1, and the independent variable x appears in the exponent, enabling it to model rapid growth or decay processes across various scientific and real-world contexts.^[1] This function is characterized by its base a, which determines whether it increases (if a > 1) or decreases (if $0 < a < 1) monotonically, and it always passes through the point (0, 1) while remaining positive for all real x.^[2] The most prominent form is the natural exponential function e^x, where e \approx 2.71828 is the base of the natural logarithm, defined as the unique positive real number satisfying \ln e = 1.^[3] Key properties of the exponential function include its differentiability and the fact that its derivative equals itself, i.e., \frac{d}{dx} e^x = e^x, making it a cornerstone for solving differential equations in fields like physics and biology.^[4] It is strictly one-to-one, possessing an inverse known as the logarithm function, which allows for solving equations involving exponents.^[5] Exponential functions exhibit proportional growth or decay, where the rate of change is directly proportional to the current value, distinguishing them from linear or polynomial functions.^[6] In applications, exponential functions model phenomena such as population growth, radioactive decay, and compound interest, where quantities change at rates dependent on their size.^[7] For instance, in finance, the formula A = P(1 + r/n)^{nt} approximates continuous compounding via the limit to Pe^{rt}, optimizing predictions for investments over time.^[8] In the sciences, they describe bacterial proliferation or nuclear half-lives, underscoring their role in quantitative analysis and forecasting.^[9]

Mathematics

Definition

In mathematics, an exponential expression is defined as a^b, where a is the base—a positive real number not equal to 1—and b is the exponent—a real number.^[10] This notation represents a multiplied by itself b times when b is a positive integer, but extends to fractional and negative real exponents, and to all real exponents via the definition a^b = e^{b \ln a} where e is the base of the natural logarithm.^[11] Exponential expressions differ fundamentally from polynomial expressions, such as x^n where the variable x serves as the base raised to a fixed non-negative integer exponent n."The Exponential Function" by Shawn A. Mousel In exponentials, the variable instead occupies the exponent position with a constant base, leading to rates of growth or decay that outpace any polynomial for bases with absolute value greater than 1 as the exponent magnitude increases.The Exponential Function A. Theorem 1 B. Example 1 A basic example is $2^3 = 8, illustrating repeated multiplication for integer exponents.Exponents This extends to real exponents, such as $2^{1/2} = \sqrt{2}, defined via the limit of rational approximations to the exponent.Tutorial 42: Exponential Functions. - West Texas A&M University

Notation

In mathematical writing, the exponential is standardly denoted using superscript notation as a^b, where a is the base and b is the exponent. For the natural exponential function with base e, the notations e^x or \exp(x) are commonly used, with the latter treating it explicitly as a function.^[12] The superscript form a^b was introduced by René Descartes in 1637 in his La Géométrie, marking a shift from earlier verbal or repeated-multiplication expressions of powers.^[13] Prior to this standardization, notations varied, with earlier forms using indices or verbal descriptions. Conventions emphasize clarity in expressions: parentheses group the base when necessary, such as (2x)^3 to mean the cube of $2x, distinct from $2x^3 = 2 \times x^3.^[14] In LaTeX for professional typesetting, the command \exp(x) is preferred for the natural exponential to avoid ambiguity with superscripts in complex formulas.^[15] In programming languages, exponentiation is typically represented by the double-asterisk operator, as in a**b for Python and JavaScript implementations.^[16] The \exp(x) notation specifically denotes the exponential with base e.

Properties

Algebraic Properties

The algebraic properties of exponential expressions provide rules for simplifying and transforming them without evaluating numerical values, building on the definition of exponents as repeated multiplication for positive integers and extending to rational and real exponents.

Product Rule

For a positive real base a > 0, a \neq 1, and real exponents m and n, the product rule states that

a^m \cdot a^n = a^{m+n}.

This property arises from the repeated multiplication interpretation: multiplying a^m ( a repeated m times) by a^n ( a repeated n times) yields a repeated m + n times./11:_Exponents_and_Polynomials/11.01:_Integer_Exponents/11.1.02:_Simplify_by_Using_the_Product_Quotient_and_Power_Rules)

Quotient Rule

Similarly, the quotient rule for the same conditions on a, m, and n (with n \neq 0) is

\frac{a^m}{a^n} = a^{m-n}.

This follows from the product rule applied inversely, as division by a^n cancels n factors of a from the m factors in the numerator.^[17]

Power Rule

The power rule specifies that

(a^m)^n = a^{mn}

under the same base and exponent conditions. Raising a^m to the nth power multiplies the exponent m by n, reflecting nested repeated multiplications./11:_Exponents_and_Polynomials/11.01:_Integer_Exponents/11.1.02:_Simplify_by_Using_the_Product_Quotient_and_Power_Rules)

Change of Base

Changing the base of an exponential expression begins with integer exponents, where a^k for positive integer k denotes a multiplied by itself k times. If the original base a can be expressed as a power of a new base c > 0, c \neq 1, such as a = c^q for integer q, then

a^k = (c^q)^k = c^{qk}

by the power rule; for example, since $4 = 2^2, it follows that $4^3 = (2^2)^3 = 2^6. This method applies when bases share an integer power relationship, allowing direct algebraic transformation without evaluation. For rational exponents p/q with integers p, q > 0, and \gcd(p, q) = 1, a^{p/q} = \sqrt{a^p} = (\sqrt{a})^p. Base changes proceed similarly if roots align with the new base c; for instance, if \sqrt{a} = c^r for rational r, then a^{p/q} = (c^r)^p = c^{rp}. Such cases rely on the power and product rules extended to roots, but require compatible bases for purely algebraic simplification.^[18] In the general case for real exponent b and bases a > 0, a \neq 1, c > 0, c \neq 1,

a^b = c^{b \log_c a},

where \log_c a is the logarithm of a to base c. This formula derives from the property that \log_c (a^b) = b \log_c a, followed by exponentiation with base c; it enables base changes for arbitrary real exponents and unrelated bases.^[19] These rules combine to simplify expressions like (2^3 \cdot 2^2) / 2^4: first apply the product rule to the numerator as $2^{3+2} = 2^5, then the quotient rule yields $2^{5-4} = 2^1 = 2./11:_Exponents_and_Polynomials/11.01:_Integer_Exponents/11.1.02:_Simplify_by_Using_the_Product_Quotient_and_Power_Rules)

Analytic Properties

The natural exponential function, denoted e^x, admits a limit-based definition given by

e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n,

where n approaches infinity through positive integers and e is the base of the natural logarithm, approximately 2.71828.^[20] This representation establishes the exponential as a continuous extension of compound growth processes and underpins its analytic behavior across the real numbers.^[21] A key analytic property is the uniqueness of the exponential function as the sole solution to the initial value problem f'(x) = f(x) with f(0) = 1. This follows from the Picard-Lindelöf theorem on the existence and uniqueness of solutions to first-order ordinary differential equations, given the Lipschitz continuity of the right-hand side.^[20]/06%3A_Differential_Equations/6.03%3A_Existence_and_Uniqueness_for_Systems_of_Linear_Differential_Equations) The exponential function b^x, for base b > 0 and b \neq 1, exhibits strict monotonicity: it is strictly increasing if b > 1 and strictly decreasing if $0 < b < 1. This property stems from the positive derivative \frac{d}{dx} b^x = b^x \ln b, which maintains the same sign as \ln b for all real x.^[20]/10%3A_Exponential_Functions/10.02%3A_Derivatives_of_Exponential_Functions) In particular, the derivative of the general exponential a^x (with a > 0, a \neq 1) is a^x \ln a, reflecting the function's self-similar scaling under differentiation up to the constant factor \ln a. For the natural base, this simplifies to \frac{d}{dx} e^x = e^x, highlighting the exponential's role as its own derivative./10%3A_Exponential_Functions/10.02%3A_Derivatives_of_Exponential_Functions)^[20]

Exponential Functions

General Form

The general form of the exponential function is given by f(x) = a^x, where a > 0, a \neq 1, and x \in [\mathbb{R}](/page/R).^[22] This equation defines a function that transforms any real input into a positive real output, establishing the domain as all real numbers \mathbb{R} and the range as the positive reals (0, \infty).^[23] The graph of f(x) = a^x for a > 1 passes through the point (0, 1) and is asymptotic to the line y = 0 as x approaches negative infinity, reflecting its behavior as a strictly increasing function that starts near zero for large negative inputs and rises without bound for positive inputs.^[2] For $0 < a < 1, the graph is a mirror image across the y-axis, decreasing from large values to approach zero asymptotically.^[23] A representative example is f(x) = 2^x, which demonstrates the characteristic rapid growth: f(0) = 1, f(1) = 2, f(3) = 8, and f(10) = 1024, with the curve steepening markedly for larger x.^[22]

Special Cases

The natural exponential function, denoted e^x, has base e \approx 2.71828, where e is defined as the limit e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n.^[24] This function arises naturally in contexts involving continuous growth and is fundamental in calculus due to its self-derivative property, \frac{d}{dx} e^x = e^x.^[25] The common exponential function, $10^x, uses base 10 and is prevalent in scientific notation, which expresses numbers as a \times 10^b where $1 \leq a < 10, facilitating the handling of extreme scales in measurements.^[26] It also underpins the pH scale in chemistry, defined as \mathrm{pH} = -\log_{10} [\mathrm{H}^+], where each unit change represents a tenfold variation in hydrogen ion concentration.^[27] The binary exponential function, $2^x, is significant in computing because binary systems inherently favor powers of 2; for instance, memory capacities like 1 KB (1024 bytes) and addressing schemes allocate space in binary increments, optimizing hardware efficiency.^[28] For any base a > 0 with a \neq 1, the exponential function a^x is bijective from the real numbers to the positive reals, ensuring its inverse, the logarithm \log_a y, is uniquely defined such that \log_a (a^x) = x and a^{\log_a y} = y for y > 0.^[29] This inverse relationship holds across all valid bases, providing a foundational tool for solving exponential equations.^[30]

Growth and Decay

Exponential Growth

Exponential growth describes a process where a quantity increases at a rate proportional to its current value, leading to accelerating expansion over time. This model is fundamental in various scientific contexts, such as population dynamics. In the continuous case, it is governed by the differential equation \frac{dy}{dt} = k y, where y(t) represents the quantity at time t, y_0 is the initial value, and k > 0 is the growth constant.^[31] The solution to this equation is y(t) = y_0 e^{kt}, which shows the quantity growing without bound as t increases, with the exponential function e^x providing the basis for this form (detailed in Special Cases).^[31] For scenarios where change occurs in discrete time steps, such as generations in biological populations, exponential growth can be approximated using a recurrence relation: y_{n+1} = r y_n, where n is the time step index and r > 1 is the growth factor. Solving this iteratively yields the closed-form expression y_n = y_0 r^n, demonstrating geometric progression that mirrors the continuous model's rapid increase.^[32] This discrete formulation serves as a practical approximation when events like reproduction happen periodically rather than continuously.^[32] A key characteristic of exponential growth is the doubling time, the period required for the quantity to double in size, which remains constant regardless of starting value. For the continuous model, this is given by T_d = \frac{\ln 2}{k}.^[33] In the discrete case with growth factor r, the doubling occurs after n steps where r^n = 2, so n = \frac{\ln 2}{\ln r}. This metric highlights the model's predictable yet explosive nature.^[34] A representative example is bacterial population growth, where each bacterium divides to produce two offspring per generation, corresponding to a discrete model with r = 2. Starting from y_0 = 1 cell, the population after n generations is y_n = 2^n, doubling with each step and reaching 1024 cells after 10 generations, illustrating the rapid proliferation under ideal conditions.^[35]

Exponential Decay

Exponential decay describes a process in which the rate of decrease of a quantity is directly proportional to the current value of that quantity, modeled by the differential equation \frac{dy}{dt} = -ky where k > 0 is the decay constant.^[36] This contrasts with exponential growth, which uses a positive rate constant in the same form.^[37] The solution to this first-order linear differential equation, assuming an initial value y(0) = y_0, is y(t) = y_0 e^{-kt}, indicating that the quantity diminishes exponentially over time t. A key characteristic of exponential decay is the half-life, t_{1/2}, defined as the time required for the quantity to reduce to half its initial value, given by the formula t_{1/2} = \frac{\ln 2}{k}.^[38] This duration remains constant regardless of the starting amount, providing a standardized measure for comparing decay processes.^[39] For instance, after one half-life, the remaining quantity is y_0 / 2; after two half-lives, it is y_0 / 4; and so on, following the pattern y(t) = y_0 \cdot 2^{-t / t_{1/2}}.^[40] In discrete-time settings, exponential decay can be approximated by the recurrence relation y_{n+1} = y_n (1 - r), where $0 < r < 1 is the fractional decay rate per time step, leading to the closed-form expression y_n = y_0 (1 - r)^n.^[41] This model is useful for scenarios where changes occur in fixed intervals, such as periodic measurements.^[42] A prominent application of exponential decay is in radioactive decay, where the number of undecayed nuclei decreases proportionally to the current number, governed by N(t) = N_0 e^{-\lambda t} with decay constant \lambda, and the half-life t_{1/2} = \ln 2 / \lambda is independent of the initial number of nuclei N_0.^[43] This constancy arises from the probabilistic nature of nuclear decay, where each nucleus has an equal chance of decaying per unit time, unaffected by the presence of others.^[39] For example, isotopes like carbon-14 exhibit this behavior, enabling reliable dating techniques based on measured half-lives.^[44]

Applications

In Natural Sciences

In physics, exponential functions model the rate of heat transfer in processes like cooling. Newton's law of cooling states that the rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature, leading to an exponential approach to equilibrium. The temperature T(t) at time t is given by

T(t) = T_a + (T_0 - T_a) e^{-kt},

where T_a is the ambient temperature, T_0 is the initial temperature, and k > 0 is a constant depending on the object's properties and environment. This law applies to convection-dominated cooling, such as a hot object in air, and has been experimentally validated for small temperature differences. In biology, exponential models describe the initial phase of population growth in unconstrained environments before resource limitations take effect. According to the Malthusian model, population size P(t) grows as P(t) = P_0 e^{rt}, where P_0 is the initial population, r is the intrinsic growth rate, and t is time; this assumes constant per capita growth without density-dependent factors.^[45] In limited environments, such as those approaching carrying capacity in logistic models, populations exhibit this exponential phase early on, as seen in bacterial cultures or unchecked animal populations before competition or predation intervenes.^[46] This framework highlights how rapid, unbounded growth can lead to overshoot in ecological systems.^[47] In chemistry, first-order reactions follow exponential decay kinetics, where the concentration of reactant [A](t) decreases over time. The rate law is \frac{d[A]}{dt} = -k[A], integrating to

[A](t) = [A]_0 e^{-kt},

with [A]_0 as the initial concentration and k the rate constant.^[48] This applies to unimolecular processes like radioactive decay or isomerization, where the half-life t_{1/2} = \frac{\ln 2}{k} is independent of initial concentration, enabling predictable modeling of reaction progress.^[49] A key application of exponential decay appears in radiometric dating, particularly carbon-14 dating for organic materials up to about 50,000 years old. Carbon-14, produced in the atmosphere, decays with a half-life of 5730 years via beta emission to nitrogen-14, following the decay equation detailed in the Exponential Decay section.^[50] By measuring the residual ^{14}\mathrm{C} ratio to stable carbon isotopes in samples, archaeologists determine age via t = \frac{1}{k} \ln \left( \frac{[^{14}\mathrm{C}]_0}{[^{14}\mathrm{C}]_t} \right), where k = \frac{\ln 2}{5730} years^{-1}.^[51] This method has revolutionized paleontology and archaeology, providing precise chronologies for prehistoric artifacts.^[52]

In Computing and Engineering

In computing, exponential time complexity arises in algorithms that exhaustively explore solution spaces, such as brute-force search for NP-hard problems like the subset sum problem, where the running time is O(2^n) for inputs of size n. This complexity class, denoted EXP, includes problems solvable in deterministic time bounded by $2^{O(n^k)} for some constant k, but brute-force approaches often achieve the baseline O(2^n) by enumerating all possible subsets.^[53] Such algorithms are impractical for large n due to the rapid growth, motivating subexponential improvements like dynamic programming or branch-and-bound techniques in specific cases.^[54] In signal processing, the exponential function e^{i \omega t} forms the basis for Fourier analysis, representing harmonic waves as complex sinusoids to decompose signals into frequency components.^[55] The Fourier transform integrates the signal against these exponentials: \hat{s}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} s(t) e^{-i \omega t} \, dt, enabling efficient computation via the fast Fourier transform (FFT) algorithm, which reduces complexity from O(n^2) to O(n \log n).^[55] This exponential kernel is fundamental in applications like audio filtering and image compression, where it models periodic phenomena in discrete-time signals.^[56] In electrical engineering, exponential decay governs the discharge of an RC circuit, where the capacitor voltage follows V(t) = V_0 e^{-t / \tau} with time constant \tau = RC.^[57] This equation describes how stored energy dissipates through the resistor, reaching approximately 37% of initial voltage V_0 after one \tau, and is derived from solving the first-order differential equation RC \frac{dV}{dt} + V = 0.^[57] Such models are essential in timing circuits, filters, and transient analysis, ensuring predictable behavior in systems like integrators and oscillators. The IEEE 754 standard influences the computation of e^x in floating-point arithmetic by defining the binary representation that approximations exploit for efficiency.^[58] One method reconstructs e^x by adjusting the exponent and mantissa fields of an IEEE 754 number, achieving a compact approximation with relative error under 0.1 for x in [-1, 1], suitable for embedded systems and numerical libraries.^[58] This leverages the format's structure—sign bit, biased exponent, and normalized mantissa—to avoid full series expansions like Taylor, balancing speed and precision in hardware implementations.^[59]

History

Early Concepts

The earliest precursors to exponential concepts emerged in ancient Babylonian mathematics around 2000 BCE, where scribes compiled extensive clay tablets containing tables of squares and cubes, representing powers such as n^2 and n^3 up to certain limits like 59 for squares and 32 for cubes. These tables, discovered at sites like Senkerah on the Euphrates, facilitated practical computations in astronomy, land measurement, and problem-solving, such as determining areas or volumes through repeated multiplications that echoed proto-exponential growth. By systematizing powers in sexagesimal notation, the Babylonians laid foundational tools for handling multiplicative sequences, though without a formal theory of exponents.^[60] In ancient Greece, around 300 BCE, Euclid advanced related ideas through his Elements, particularly in Book VIII, which explores continued proportions among natural numbers—equivalent to modern geometric progressions where each term is obtained by multiplying the previous by a fixed ratio. Euclid defined these sequences rigorously, proving properties like the constancy of ratios between consecutive terms and applications to sums of such series, building on earlier notions of proportion from Book V. This framework connected ratios to multiplicative scaling, providing a geometric basis for exponential-like relations without algebraic notation for powers.^[61] Indian mathematics in the 7th century CE saw further development with Brahmagupta's Brahmasphuṭasiddhānta (628 CE), which introduced rules for arithmetic operations involving negative quantities (termed "debts") and zero, enabling computations that implicitly supported negative exponents as reciprocals of positive powers. Brahmagupta specified that the product of a positive and negative number yields a negative, and he handled divisions leading to fractional results akin to negative powers, such as treating a^{-n} through inverse operations on positives. These rules, applied in solving quadratic equations with squared terms, marked a significant step toward formalizing exponential expressions with signed exponents.^[62] A notable demonstration of exponential scale in the Hellenistic period appears in Archimedes' The Sand Reckoner (c. 216 BCE), where he devised a system to enumerate extraordinarily large numbers, estimating the grains of sand needed to fill the universe by iterating powers like $10^{8 \times 10^{63}} through nested multiplications. Addressed to King Gelon, this work extended Greek numeral systems beyond the myriad (10,000) by defining orders of magnitude via repeated exponentiation, illustrating the conceptual power of exponentials for bounding infinite-like quantities in cosmology and geometry.^[63]

Modern Developments

In the early 17th century, John Napier introduced logarithms in his 1614 treatise Mirifici Logarithmorum Canonis Descriptio, which provided tables for simplifying multiplications and divisions through addition and subtraction; from a modern perspective, these logarithms represent the inverse operation to exponentiation with a constant base, bridging geometric progressions to exponential growth concepts.^[64] Napier's approach, motivated by astronomical calculations, effectively encoded exponential scaling in a tabular form, influencing subsequent mathematical analysis. Later that century, John Wallis formulated an infinite product expression for π/2 in his 1655 work Arithmetica Infinitorum, given by \frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4n^2}{4n^2 - 1}, which demonstrated the power of infinite products and foreshadowed their application in representing exponential and trigonometric functions. The 18th century marked a pivotal advancement with Leonhard Euler's systematic treatment of the exponential function in his 1748 publication Introductio in Analysin Infinitorum. There, Euler defined the constant [e](/page/E!) \approx 2.71828 as the base arising from the limit \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n, establishing it as the foundation for natural growth and decay processes in calculus.^[65] Euler further standardized the notation \exp(x) to denote [e](/page/E!)^x, promoting clarity in expressing continuous compounding and differential equations, which integrated exponentials deeply into analytic methods.^[66] A cornerstone of these developments is Euler's formula, succinctly stated as e^{i\pi} + 1 = 0, which elegantly unites the exponential function with imaginary numbers and the constants e, i, \pi, and 1; derived through series expansions, it reveals the periodic nature of complex exponentials as rotations in the plane.^[66] By the 19th century, Bernhard Riemann extended these ideas into complex analysis with his 1851 doctoral thesis Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse, where he laid the foundations for the theory of functions of a complex variable, exploring analytic functions such as the exponential and introducing the concept of Riemann surfaces to handle multi-valued functions, including the complex logarithm (the inverse of the exponential), resolving branch points to ensure single-valued representations on these surfaces.^[67] Riemann's framework unified exponential mappings with conformal geometry, enabling rigorous study of analytic continuation and influencing modern function theory.

References

[1]
Calculus I - Exponential Functions - Pauls Online Math Notes
Nov 16, 2022 · An exponential function is f(x)=bx, where b>0 and b≠1. It always takes the value of 1 at x=0 and is always positive.
[2]
[PDF] Exponential Functions - Math
An exponential function is defined as f(x) = ax, where 'a' is the base, which must be a positive number. The variable 'x' is the exponent.
[3]
[PDF] 1 Definition and Properties of the Exp Function
Number e. Definition 1. The number e is defined by ln e = 1 i.e., the unique number at which ln x = 1.
[4]
9.3 The exponential function
The natural exponential function, exp(x), is the inverse of ln(x), with domain all real numbers and range (0,∞). It's often written as ex.
[5]
Properties of the Exponential Function - Ximera - Xronos
The exponential function is one-to-one, that is it passes the horizontal line test. This means that there exists an inverse function which we call a logarithm.
[6]
Exponential Functions
The key property of exponential functions is that the rate of growth (or decay) is proportional to how much is already there.<|control11|><|separator|>
[7]
Algebra - Applications - Pauls Online Math Notes
Nov 16, 2022 · In this section we will look at a couple of applications of exponential functions and an application of logarithms.
[8]
5.7 Applications of Exponential and Logarithmic Functions
Equations 5.2 and 5.3 both use exponential functions to describe the growth of an investment. It turns out, the same principles which govern compound interest ...
[9]
Exponential Functions
The common applications of the exponential funciton range from population modeling, to tracking drug levels in the blood stream, to using carbon dating to ...
[10]
The exponential function - Math Insight
Fixing b=e, we can write the exponential functions as f(x)=ekx. (The applet understands the value of e, so you can type ...
[11]
A History of Mathematical Notations/Volume 1 - Wikisource
Dec 12, 2023 · FLORIAN CAJORI, Ph.D. Professor of the History of Mathematics University of California. Volume I. NOTATIONS IN ELEMENTARY MATHEMATICS. The Open ...
[12]
Rules of Exponents
One of the rules of exponential notation is that the exponent relates only to the value immediately to its left.
[13]
Why does the exp(x) not work in Latex? - TeX
Sep 5, 2018 · The ex notation is useful where x is something nice and small, like, well: ... So LaTeX supports both, \exp(x) for exp(x) and e^{x} for ex.Exponential function - equations - LaTeX Stack Exchangee^{...} vs \exp(...) in display mode - LaTeX Stack ExchangeMore results from tex.stackexchange.com
[14]
Exponentiation (**) - JavaScript - MDN Web Docs - Mozilla
Jul 8, 2025 · Note that some programming languages use the caret symbol ^ for exponentiation, but JavaScript uses that symbol for the bitwise XOR operator.
[15]
5.3.1: Properties of Exponents: Product, Quotient and Power Rules
The quotient rule for exponents allows us to simplify an expression that divides two numbers with the same base but different exponents.
[16]
Algebra - Rational Exponents - Pauls Online Math Notes
Nov 16, 2022 · In this section we will define what we mean by a rational exponent and extend the properties from the previous section to rational exponents ...Missing: base | Show results with:base
[17]
Change of Base Formula for Exponents - Math Stack Exchange
Jun 21, 2019 · Most exponent rules have a corresponding log rule and vice versa. For example, abac=ab+c and loga(bc)=loga(b)+loga(c).Is there a "Exponential Form" of the "Logarithmic Change of Base"?Intuition behind logarithm change of base - Math Stack ExchangeMore results from math.stackexchange.com
[18]
Exponential Function -- from Wolfram MathWorld
The most general form of an exponential function is a power-law function of the form f(x)=ab^(cx+d), where a, c, and d are real numbers, b is a positive real ...
[19]
e -- from Wolfram MathWorld
e is the most important constant in mathematics since it appears in myriad mathematical contexts involving limits and derivatives.
[20]
Algebra - Exponential Functions - Pauls Online Math Notes
Nov 16, 2022 · If b b is any number such that b>0 b > 0 and b≠1 b ≠ 1 then an exponential function is a function in the form,. f(x)=bx f ( x ) = b x.
[21]
4.1 - Exponential Functions and Their Graphs
Graphs of Exponential Functions · The graph passes through the point (0,1) · The domain is all real numbers · The range is y>0. · The graph is decreasing · The graph ...
[22]
The Number e and the Exponential Function - Galileo
The Exponential Function ex. Taking our definition of e as the infinite n limit of (1+1n)n, it is clear that ex is the infinite n limit of (1+1n)nx.
[23]
2.9: Limit of Exponential Functions and Logarithmic Functions
Dec 20, 2020 · Since functions involving base e arise often in applications, we call the function f ⁡ ( x ) = e x the natural exponential function. Not only is ...
[24]
[PDF] Powers of 10 & Scientific Notation
Since our number system is “base-10” (based on powers of 10), it is far more convenient to write very large and very small numbers in a special exponential.
[25]
8.6: The pH Concept - Chemistry LibreTexts
Jun 9, 2019 · Notice that when [ H ⁡ A + ] is written in scientific notation and the coefficient is 1, the pH is simply the exponent with the sign changed.
[26]
[PDF] 1 Powers of two or exponentials base two - NJIT
Aug 31, 2016 · A byte is the minimal amount of binary information that can be stored into the memory of a computer and it is denoted by a capital case B.<|control11|><|separator|>
[27]
Logarithm as Inverse of Exponential - Maple Help - Maplesoft
The functions and are inverses of each other. The domain of the logarithm base is all positive numbers. The range of the logarithm base is all real numbers.
[28]
Inverse Functions and Logarithms
This inverse should be familiar: the number x x for which ax=y a x = y is called the logarithm of y y base a a , written logay, log a ⁡ y , so the inverse of ...<|control11|><|separator|>
[29]
Exponential growth and decay: a differential equation - Math Insight
### Summary of Exponential Growth from Math Insight
[30]
Exponential growth and decay modeled by discrete dynamical ...
Overview of exponential growth and decay in discrete time. Exploration of their qualitative properties as well as solutions to the dynamical system.
[31]
[PDF] 2.8 | Exponential Growth and Decay
Introduction to Differential Equations. Rule: Exponential Growth Model. Systems that exhibit exponential growth increase according to the mathematical model.
[32]
Doubling time and half-life of exponential growth and decay
Such exponential growth or decay can be characterized by the time it takes for the population size to double or shrink in half. For exponential growth, we can ...
[33]
[PDF] 1.5 Discrete-Time Dynamical Systems
A Discrete-Time Dynamical System for a Bacterial Population. Recall the ... A population of bacteria doubles every hour, but 1.0 × 106 individuals are ...
[34]
[PDF] 5.9 Exponential Growth and Decay
The doubling time for exponential growth and the half-life for exponential decay are both equal to (In 2)/k. For use in carbon dating: The decay constant of C14 ...
[35]
Exponential growth and decay: a differential equation - Math Insight
If the constant k is positive it has exponential growth and if k is negative then it has exponential decay.
[36]
[PDF] Math 1131 Applications: Exponential Growth/Decay
A modern experimental method to measure half-life is scintillation counting. Half-life can be related to “k” in the exponential decay formula y(t) = Cekt with ...
[37]
Radioactive Half-Life - HyperPhysics Concepts
If there are N radioactive nuclei at some time t, then the number ΔN which would decay in any given time interval Δt would be proportional to N: where λ is a ...
[38]
Logarithmic and exponential models - Pre-Calculus
When an exponential function is used to study a problem arising from an application, it is called an exponential model. If an increasing exponential function ( ...<|control11|><|separator|>
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[PDF] Math 1311 Section 4.1 Exponential Growth and Decay
Decay: For an exponential function with discrete (yearly, monthly, etc.) percentage decay rate. 𝑟 as a decimal, the decay factor 𝒂 = 𝟏 – 𝒓. Example 5: A ...
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[PDF] Discrete and continuous models; Linear and exponential growth
In general, the difference equation for the discrete exponential model takes the form. (1). ∆P = rP and the general solution is: P(t) = P0(1 + r)t. where P0 ...
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[PDF] Exponential Decay - Purdue Math
Apr 28, 2021 · Since the decay rate of a radioactive isotope is characterized by half-life, the constant k in the decay model must have some connection to half ...
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[PDF] Radioactive Decay Laws - isnap
λ≡decay constant; a natural constant for each radioactive element. Half life: t. 1/2. = ln2/λ exponential decay with time! At half life 50% of the activity is ...
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[PDF] 1.5 Malthusian Growth - Purdue Math
Feb 16, 2007 · Assuming an exponential growth law, determine the time it takes for the culture to contain 106 cells. 4. At time t, the population P(t) of a ...
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Population Growth Models - Biological Principles
With exponential population growth, the population growth rate r was constant, but with the addition of a carrying capacity imposed by the environment, ...
[45]
Population Growth and Regulation – Introductory Biology
The early pattern of accelerating population size is called exponential growth. The best example of exponential growth in organisms is seen in bacteria.
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First order reactions
That is, the amount of (CH3)3CBr decays away exponentially with time from its initial value at t = 0, [(CH3)3CBr]0. The decay constant, k, is the rate constant.
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[PDF] Chapter 14 Chemical Kinetics - MSU chemistry
The half-life of a first-order reaction depends only on the rate constant: ▫ independent of the initial reactant concentration. [R] t =[R].
[48]
Radioactive Decay - Education - Stable Isotopes NOAA GML
14C has a half-life of 5,730 years. In other words, after 5,730 years, only half of the original amount of 14C remains in a sample of organic material.
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How Do You Know the Age of Fossils and Other Old Things? | NIST
Feb 24, 2025 · As carbon-14 decays, with a half-life of about 5,730 years, it becomes nitrogen-14. Using this clock, they have dated bones, campfires and other ...
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DOE Explains...Isotopes - Department of Energy
Carbon-14 is unstable and undergoes radioactive decay with a half-life of about 5,730 years (meaning that after 5,730 years half of the material will ...
[51]
[PDF] Exact Exponential Algorithms for Clustering Problems - arXiv
Aug 14, 2022 · For many problems, the running time of O∗(2n) is often achievable by a brute-force enumeration of all the solutions. However, for many NP ...
[52]
On Exponential-time Hypotheses, Derandomization, and Circuit ...
Apr 20, 2023 · ... linear space (by a brute-force algorithm) or in time (by Shoup's [53] algorithm). Fix a bijection between and (i.e., maps any string in ...
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[PDF] Optimally Repurposing Existing Algorithms to Obtain Exponential ...
Jun 27, 2023 · Note that brute(1) = 2, i.e., in the exact setting, this recovers the standard brute-force algorithm running in time O∗(2n). We compare ...
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[PDF] A guide on spectral methods applied to discrete data Part I - arXiv
Dec 22, 2016 · Given a real-valued time-dependent signal s(t), the Fourier transform can be calculated by. S (ω) = 1. √. 2π. Z ∞. −∞ s(t) · e. −iω·t dt. (1).
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[PDF] Convolutional discrete Fourier transform method for calculating ...
Dec 14, 2021 · Discrete Fast Fourier transform is typically used to correlate atomic microscopic structure and dynamics with measurable scatter- ing cross ...<|separator|>
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[PDF] Experiment 1: RC Circuits
The potential will decrease with time according to the relation: V(t) = V0e. −t /τ , where τ = RC. (1). V0 represents the voltage at time t = 0, and τ ...
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http://users.physics.ucsd.edu/2009/Fall/physics2cl/documents/Experiment1.pdf
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https://ieeexplore.ieee.org/document/6790798
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Babylonian mathematics - University of St Andrews
The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years.
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[PDF] Euclid English version
The eighth book of the Elements deals with continued proportions of natural numbers (in today's language, with geometric sequences). For example, the three ...
[61]
Brahmagupta (598 - 670) - Biography - University of St Andrews
Brahmagupta, whose father was Jisnugupta, wrote important works on mathematics and astronomy. In particular he wrote Brahmasphutasiddhanta Ⓣ. (Correctly ...Missing: powers | Show results with:powers
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Archimedes - Biography - University of St Andrews
The Sandreckoner is a remarkable work in which Archimedes proposes a number system capable of expressing numbers up to 8 × 1 0 63 8 \times 10^{63} 8×1063 ...
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John Napier (1550 - Biography - MacTutor History of Mathematics
Napier also found exponential expressions for trigonometric functions, and introduced the decimal notation for fractions. Much of Napier's work on logarithms ...
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The number e - MacTutor History of Mathematics
It would be fair to say that Johann Bernoulli began the study of the calculus of the exponential function in 1697 when he published Principia calculi ...
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Leonhard Euler (1707 - 1783) - Biography - University of St Andrews
Leonhard Euler was a Swiss mathematician who made enormous contributions to a wide range of mathematics and physics including analytic geometry, trigonometry, ...