Name: solving population growth with logarithms
Uploaded: 2025-11-26T05:43:38.000Z
Duration: 1 min 29 s
Channel: JoVE

在河狸栖息地广阔的森林区域，研究人员仔细追踪河狸种群随时间的增长情况。目标是确定人口数量达到特定规模所需的年数。该种群遵循基于随时间重复增长的指数模型。其数量等于初始种群数量乘以10的（增长率乘以年数）次方。增长率表示种群每年增长的速度。开始计算时，研究人员将目标群体数值代入方程。将等式两边同时除以初始种群数量，可得到种群增长的倍数。然后对等式进行变形，使 10 的某个指数次幂等于该倍数。由于对数和指数是互为逆运算，对方程两边取对数可使变量分离。接着应用幂的对数法则，将指数移至下方，从而使方程转化为可求解的线性形式。指数现在明显表现为常数与年数的乘积。将对数值除以该常数，即可估算出种群数量达到预期最终规模所需的大致年数。

指数与对数方程：问题求解

Exponential Equations with Logarithms: Problem Solving

&#22312;&#29983;&#24577;&#23398;&#30740;&#31350;&#20013;&#65292;&#25351;&#25968;&#27169;&#22411;&#36890;&#24120;&#29992;&#20110;&#39044;&#27979;&#22312;&#26377;&#21033;&#26465;&#20214;&#19979;&#31181;&#32676;&#38543;&#26102;&#38388;&#30340;&#21464;&#21270;&#36235;&#21183;&#12290;&#27492;&#31867;&#27169;&#22411;&#20551;&#35774;&#31181;&#32676;&#30340;&#22686;&#38271;&#29575;&#19982;&#20854;&#24403;&#21069;&#35268;&#27169;&#25104;&#27491;&#27604;&#65292;&#20174;&#32780;&#20135;&#29983;&#36830;&#32493;&#19988;&#22797;&#21512;&#30340;&#22686;&#38271;&#36807;&#31243;&#12290;
&#35813;&#27169;&#22411;&#23558;&#31181;&#32676;&#35268;&#27169;&#34920;&#31034;&#20026;&#26102;&#38388;&#30340;&#20989;&#25968;&#65292;&#23558;&#21021;&#22987;&#31181;&#32676;&#25968;&#37327;&#19982;&#19968;&#20010;&#21253;&#21547;&#22686;&#38271;&#29575;&#21644;&#26102;&#38388;&#30340;&#25351;&#25968;&#22686;&#38271;&#22240;&#23376;&#30456;&#32467;&#21512;&#12290;&#20026;&#20102;&#20272;&#31639;&#31181;&#32676;&#36798;&#21040;&#29305;&#23450;&#35268;&#27169;&#25152;&#38656;&#30340;&#26102;&#38388;&#65292;&#30740;&#31350;&#20154;&#21592;&#23558;&#30446;&#26631;&#31181;&#32676;&#20540;&#20195;&#20837;&#27169;&#22411;&#65292;&#24182;&#23558;&#20854;&#38500;&#20197;&#21021;&#22987;&#20540;&#12290;&#27492;&#36807;&#31243;&#24471;&#21040;&#30340;&#22686;&#38271;&#22240;&#23376;&#21453;&#26144;&#20102;&#31181;&#32676;&#25968;&#37327;&#30456;&#36739;&#20110;&#21021;&#22987;&#35268;&#27169;&#30340;&#20493;&#25968;&#12290;
&#30001;&#20110;&#26102;&#38388;&#21464;&#37327;&#20986;&#29616;&#22312;&#22686;&#38271;&#34920;&#36798;&#24335;&#30340;&#25351;&#25968;&#37096;&#20998;&#65292;&#27714;&#35299;&#25152;&#38656;&#26102;&#38388;&#38656;&#36890;&#36807;&#36870;&#21521;&#25351;&#25968;&#36816;&#31639;&#27714;&#35299;&#12290;&#36825;&#19968;&#36807;&#31243;&#20381;&#36182;&#23545;&#25968;&#26041;&#27861;&#65292;&#21363;&#21033;&#29992;&#23545;&#25968;&#30340;&#24615;&#36136;&#65292;&#23558;&#26102;&#38388;&#29992;&#24050;&#30693;&#37327;&#65288;&#22914;&#21021;&#22987;&#31181;&#32676;&#12289;&#26368;&#32456;&#31181;&#32676;&#20197;&#21450;&#22686;&#38271;&#29575;&#65289;&#34920;&#31034;&#20986;&#26469;&#12290;&#36890;&#36807;&#22522;&#20110;&#23545;&#25968;&#30340;&#36923;&#36753;&#37325;&#26500;&#27169;&#22411;&#65292;&#26102;&#38388;&#21464;&#37327;&#21487;&#35299;&#26512;&#27714;&#24471;&#65292;&#20174;&#32780;&#25581;&#31034;&#22312;&#25345;&#32493;&#22686;&#38271;&#26465;&#20214;&#19979;&#31181;&#32676;&#36798;&#21040;&#30446;&#26631;&#35268;&#27169;&#25152;&#38656;&#30340;&#20855;&#20307;&#26102;&#38271;&#12290;
&#36825;&#19968;&#20998;&#26512;&#36807;&#31243;&#31361;&#20986;&#20102;&#23545;&#25968;&#22312;&#27714;&#35299;&#25351;&#25968;&#26041;&#31243;&#20013;&#30340;&#20851;&#38190;&#20316;&#29992;&#65292;&#20351;&#30740;&#31350;&#32773;&#33021;&#22815;&#23450;&#37327;&#20272;&#31639;&#31181;&#32676;&#22686;&#38271;&#33267;&#26399;&#26395;&#35268;&#27169;&#25152;&#38656;&#30340;&#26102;&#38388;&#12290;&#23545;&#25968;&#26041;&#27861;&#22240;&#27492;&#25104;&#20026;&#31181;&#32676;&#24314;&#27169;&#19982;&#36164;&#28304;&#31649;&#29702;&#20013;&#30340;&#22522;&#30784;&#25968;&#23398;&#24037;&#20855;&#12290;

exponential-equations-with-logarithms-problem-solving

Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Exponential and Logarithmic Functions

指数与对数函数

KEY TERMS AND CONCEPTS

关键术语与概念

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 population reaches a target size, the target value is substituted into the equation and divided by the starting population. This shows the growth factor, or how many times the population has multiplied.

Because the time value is inside the exponential expression, the equation must be reversed to solve for years. Logarithmic reasoning makes that possible. It rewrites the known values, such as the initial and final population sizes and the growth rate, so time can be calculated directly. This method shows how logarithms are used to estimate how long a population will take to reach a desired size under steady growth conditions.

These steps are useful in population modeling and resource management. They also show why logarithms are an important tool for working with exponential equations in ecology.

Solving Population Growth with Logarithms

solving-population-growth-with-logarithms

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...

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

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

Learn how logarithms solve exponential population growth problems by finding the time needed for a population to reach a target size in ecology.

Solving Population Growth with Logarithms | 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

Solving Population Growth with Logarithms

Description

Transcript