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

指数関数は、1 に等しくない正の底を実数指数に上げることによって定義されます。その一般的な形式は、初期値に指数に上げられた底数を掛けたものです。基数が 1 より大きい場合、関数は成長をモデル化します。0 から 1 の間にある場合、包括的でないときは、減衰をモデル化します。たとえば、注射後に血流中の初期濃度が1ミリリットルあたり30マイクログラムに達し、毎時間濃度の75%を保持する薬を考えてみましょう。この場合、指数関数は 30 に 0.75 を掛けて指数 t に上げ、経過した時間を表します。1時間後、濃度は1ミリリットルあたり22.5マイクログラムに低下します。2時間後、1ミリリットルあたり16.9マイクログラムになり、パターンが続きます。グラフでは、曲線は最初は急速に下がり、その後徐々に平坦になり、ゼロで水平漸近線に近づきます。時間は負にできないので、領域は負でない実数です。濃度がゼロに達することはないため、範囲には正の値のみが含まれます。

指数関数入門

&#25351;&#25968;&#38306;&#25968;&#12399;&#12289;&#22793;&#21270;&#29575;&#12364;&#29694;&#22312;&#12398;&#20516;&#12395;&#27604;&#20363;&#12377;&#12427;&#21205;&#30340;&#12503;&#12525;&#12475;&#12473;&#12434;&#12514;&#12487;&#12523;&#21270;&#12377;&#12427;&#19978;&#12391;&#22522;&#26412;&#30340;&#12394;&#38306;&#25968;&#12391;&#12377;&#12290;f(x) = b^x&#65288;b&#12399;1&#20197;&#22806;&#12398;&#27491;&#12398;&#23450;&#25968;&#65289;&#12391;&#23450;&#32681;&#12373;&#12428;&#12289;&#24213;b&#12364;1&#12424;&#12426;&#22823;&#12365;&#12356;&#12363;&#23567;&#12373;&#12356;&#12363;&#12395;&#24540;&#12376;&#12390;&#12289;&#22679;&#21152;&#12392;&#28187;&#23569;&#12398;&#12503;&#12525;&#12475;&#12473;&#12434;&#35352;&#36848;&#12377;&#12427;&#12383;&#12417;&#12398;&#22522;&#30990;&#12392;&#12394;&#12426;&#12414;&#12377;&#12290;&#25351;&#25968;&#38306;&#25968;&#12514;&#12487;&#12523;&#12399;&#12289;&#22793;&#21270;&#12364;&#29694;&#22312;&#12398;&#37327;&#12395;&#27604;&#20363;&#12377;&#12427;&#36895;&#24230;&#12391;&#29983;&#12376;&#12427;&#29366;&#27841;&#12434;&#34920;&#12375;&#12414;&#12377;&#12290;&#12371;&#12428;&#12395;&#12399;&#12289;&#32048;&#33740;&#12398;&#22679;&#27542;&#12289;&#25918;&#23556;&#24615;&#29289;&#36074;&#12398;&#23849;&#22730;&#12289;&#36039;&#29987;&#12398;&#28187;&#20385;&#20767;&#21364;&#12394;&#12393;&#12398;&#29694;&#35937;&#12364;&#21547;&#12414;&#12428;&#12414;&#12377;&#12290;&#25351;&#25968;&#38306;&#25968;&#12514;&#12487;&#12523;&#12398;&#19968;&#33324;&#24418;&#12399;&#27425;&#12398;&#12424;&#12358;&#12395;&#34920;&#12373;&#12428;&#12414;&#12377;&#12290;<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>&#12371;&#12371;&#12391;&#12289;A_0 &#12399;&#21021;&#26399;&#20516;&#12289;&#24213; b &#12399;&#25104;&#38263;&#20418;&#25968;&#65288;b &gt; 1&#65289;&#12414;&#12383;&#12399;&#28187;&#34928;&#20418;&#25968;&#65288;0 &lt; b &lt; 1&#65289;&#12434;&#31034;&#12375;&#12289;t &#12399;&#26178;&#38291;&#12414;&#12383;&#12399;&#12381;&#12398;&#20182;&#12398;&#36899;&#32154;&#22793;&#25968;&#12434;&#34920;&#12375;&#12414;&#12377;&#12290;&#24179;&#34892;&#31227;&#21205;&#12289;&#21453;&#23556;&#12289;&#25313;&#22823;&#12539;&#32302;&#23567;&#12392;&#12356;&#12387;&#12383;&#22793;&#25563;&#12399;&#25351;&#25968;&#38306;&#25968;&#12398;&#12464;&#12521;&#12501;&#12398;&#24418;&#12434;&#22793;&#12360;&#12414;&#12377;&#12364;&#12289;&#22522;&#26412;&#30340;&#12394;&#24418;&#29366;&#12420;&#28472;&#36817;&#32218;&#12398;&#24615;&#36074;&#12399;&#20445;&#12383;&#12428;&#12414;&#12377;&#12290;&#12373;&#12425;&#12395;&#12289;&#25351;&#25968;&#38306;&#25968;&#12398;&#21336;&#23556;&#24615;&#12395;&#12424;&#12426;&#12289;b^x = b^y &#12398;&#22580;&#21512;&#12289;x = y &#12364;&#25104;&#12426;&#31435;&#12385;&#12414;&#12377;&#12290;&#12371;&#12398;&#24615;&#36074;&#12434;&#21033;&#29992;&#12375;&#12390;&#12289;&#21516;&#12376;&#24213;&#12395;&#12381;&#12429;&#12360;&#12390;&#27604;&#36611;&#12377;&#12427;&#12371;&#12392;&#12391;&#25351;&#25968;&#12434;&#21547;&#12416;&#26041;&#31243;&#24335;&#12434;&#35299;&#12367;&#12371;&#12392;&#12364;&#12391;&#12365;&#12414;&#12377;&#12290;

Education

JoVE Core

Math Fundamentals

数学の基礎

Unit

ユニット

Exponential and Logarithmic Functions

指数関数と対数関数

KEY TERMS AND CONCEPTS

主要用語と概念

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

Description

Transcript

Tags