Name: exponential growth and decay models
Uploaded: 2025-11-26T05:43:46.000Z
Duration: 1 min 29 s
Channel: JoVE

Una función exponencial se define por una base positiva, no igual a uno, elevada a un exponente de número real.Su forma general es un valor inicial multiplicado por la base elevada a un exponente.Cuando la base es mayor que uno, la función modela el crecimiento. Cuando está entre cero y uno, de manera no inclusiva, modela la decadencia.Por ejemplo, considere un medicamento que alcanza una concentración inicial de 30 microgramos por mililitro en el torrente sanguíneo después de la inyección y retiene el 75 por ciento de su concentración cada hora.En este caso, la función exponencial es 30 multiplicado por 0,75 elevado al exponente t, que representa las horas transcurridas.Después de una hora, la concentración cae a 22,5 microgramos por mililitro. Después de dos horas, se convierte en 16.9 microgramos por mililitro y el patrón continúa.En un gráfico, la curva cae rápidamente al principio, luego se aplana gradualmente, acercándose a una asíntota horizontal en cero.Dado que el tiempo no puede ser negativo, el dominio son los números reales no negativos. Debido a que la concentración nunca llega a cero, el rango incluye solo valores positivos.

introducción a las funciones exponenciales

Las funciones exponenciales son fundamentales en el modelado de procesos din&#225;micos en los que la tasa de cambio es proporcional al valor actual. Definidas por f(x) = b^x, donde b es una constante positiva distinta de uno, constituyen la base para describir procesos de crecimiento y disminuci&#243;n seg&#250;n si la base b es mayor o menor que uno.Los modelos exponenciales describen situaciones en las que el cambio se produce a un ritmo proporcional a la cantidad actual. Estos modelos incluyen fen&#243;menos como la proliferaci&#243;n bacteriana, la desintegraci&#243;n de sustancias radiactivas y la depreciaci&#243;n de activos. La forma general de un modelo exponencial puede expresarse como:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1920/84;' src='https://cloudfront.jove.com/files/ftp_upload/40061/40061_Equation_1_editor_40061.svg' alt='40061_Equation_1.svg' /></figure>Aqu&#237;, A_0 es la cantidad inicial, b es la base que indica el factor de crecimiento (b &gt; 1) o de disminuci&#243;n (0 &lt; b &lt; 1), y t representa el tiempo u otra variable continua.Las transformaciones, tales como traslaciones, reflexiones y estiramientos, modifican la representaci&#243;n gr&#225;fica de las funciones exponenciales, pero conservan su forma fundamental y su comportamiento asint&#243;tico. Adem&#225;s, la Propiedad uno a uno de las funciones exponenciales garantiza que, si b^x = b^y, entonces x = y, lo que permite resolver ecuaciones con exponentes mediante la comparaci&#243;n de bases.

Introducción a las funciones exponenciales

Las funciones exponenciales son fundamentales en el modelado de procesos din&#225;micos en los que la tasa de cambio es proporcional al valor actual. ...

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JoVE Core

Math Fundamentals

Fundamentos de matemáticas

Unit

Unidad

Exponential and Logarithmic Functions

Funciones exponenciales y logarítmicas

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

Introduction to Exponential Functions

exponential growth and decay models

Exponential functions model growth and decay when the rate of change depends on the current value. They are written as f(x) = b^x, where b is a positive constant and not equal to 1. When the base b is greater than 1, the function shows growth. When 0 &lt; b &lt; 1, the function shows decay.

These models are useful in many real situations. They can describe bacterial proliferation, radioactive substance decay, and the depreciation of assets. In each case, the change is proportional to the current amount. A general exponential model can be written as A₀b^t, where A₀ is the initial amount, b is the growth or decay factor, and t is time or another continuous variable.

Exponential graphs can also be changed with transformations. Translations, reflections, and stretches shift or reshape the graph, but the basic curve and asymptotic behavior stay the same. Exponential functions also have the One-to-One Property. If b^x = b^y, then x = y, which lets you solve equations with exponents by comparing the bases.

Exponential Growth and Decay Models

introduction to exponential functions

Exponential functions model growth and decay when the rate of change depends on the current value. They are written as f(x) = b^x, where b is a positive constant and not equal to 1. When the base b is greater than 1, the function shows growth. When 0 &lt; b &lt; 1, the function shows decay. These models are useful in many real situations. They can describe bacterial proliferation, radioactive substance decay, and the depreciation of assets. In each case, the change is proportional to the...

exponential functions with base e

Exponential Growth and Decay in Real Life

Exponential functions with base e model continuous growth and decay in real life. The constant e is about 2.718. It appears naturally when change depends on the current value. A positive exponent shows continuous growth, while a negative exponent shows continuous decay. These functions are useful when change happens smoothly over time instead of in steps. One example of exponential decay is a hot beverage cooling down. The temperature drops quickly at first. As it gets closer to room...

introduction to logarithmic functions

Logarithms as Exponents Reversed

Logarithmic functions are the inverse of exponential functions. They are used to solve for exponents. In the general form y = log_a(x), the base a must be greater than 0 and not equal to 1. A logarithm tells you the power needed to get a given number. In other words, y = log_a(x) gives the exponent that the base a must be raised to in order to obtain x. The function is defined only for x &gt; 0, but its range includes all real numbers. The graphs of logarithmic and exponential functions are...

laws of logarithms i

Logarithm Rules for Product and Quotient

Logarithm rules help simplify expressions that use powers, multiplication, and division. A logarithm is the inverse of exponentiation, so it shows how many times a base must be raised to reach a number. For base 10, the common logarithm is written as log. If 10^n = x, then log(x) = n. These rules are useful because they turn harder calculations into simpler steps. The product law says that the logarithm of a product equals the sum of the logarithms. For example, log(100) can be rewritten as...

laws of logarithms ii

Power Law and Change-of-Base Formula

Logarithmic laws help simplify exponential expressions and make them easier to evaluate. Two key tools are the power law and the change-of-base formula. These rules are useful in math and in applied settings where powers and repeated multiplication appear often. The power law says that the logarithm of a number raised to an exponent equals the exponent multiplied by the logarithm of the number. This rule is helpful when solving equations that involve growth over time, such as compound interest.

exponential equations with logarithms: problem solving

exponential-equations-with-logarithms-problem-solving

Solving Population Growth with Logarithms

Logarithms help solve population growth problems with exponential models. In ecology, these models are used to predict how a population changes over time when conditions stay favorable. The growth rate is proportional to the current population, so the number increases in a continuous and compounding way. The model writes population as a function of time. It combines the initial population with a growth factor raised to an exponent that includes the growth rate and the time. To find when a...

applications of logarithms

Logarithms in Sound and Decibels

Logarithms help describe sound intensity with the decibel scale. They are useful when a quantity changes over a very wide range. In acoustics, the decibel (dB) scale turns sound levels into numbers that are easier to compare and use. Sound can range from a faint whisper to the roar of a jet engine. A linear scale would make those differences hard to manage. The decibel scale is a logarithmic scale, which means it measures sound intensity level relative to a reference value. For air, that...

exponential equations for modeling growth

exponential-equations-for-modeling-growth

Exponential Growth in Population Models

Exponential growth in population models describes rapid, multiplicative change. It is useful when a population grows by the same factor at regular intervals. This makes it a strong fit for early-stage growth in natural systems. A bacterial culture that doubles every three hours can be modeled with an exponential equation. The model is n(t)=n0·2^(t/3), where n(t) is the population at time t and n0 is the starting population. The base 2 matches the doubling pattern. A more general exponential...

key terms and concepts

Explore exponential growth and decay models, graph transformations, real-world examples like bacteria and radioactive decay, and the one-to-one property.

Exponential Growth and Decay Models | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Exponential Growth and Decay Models

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