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

비버 서식지가 넓은 숲이 우거진 지역에서 연구원은 시간이 지남에 따라 비버 개체수가 어떻게 증가하는지 주의 깊게 추적합니다.목표는 인구가 특정 크기에 도달하는 데 필요한 연수를 결정하는 것입니다.인구는 시간이 지남에 따라 반복되는 성장을 기반으로 하는 기하급수적인 모델을 따릅니다. 이는 초기 인구에 10을 곱한 값에 성장률에 연수를 곱한 값과 같습니다. 성장률은 인구가 매년 얼마나 빨리 증가하는지를 보여줍니다.계산을 시작하기 위해 연구원은 목표 모집단 값을 방정식에 대입합니다.초기 인구로 양쪽을 나누면 인구가 증가한 요인이 산출됩니다. 그런 다음 지수로 올린 10이 해당 인수와 같도록 방정식을 재배열합니다.로그와 지수는 역연산이므로 양쪽의 로그를 취하면 변수가 분리됩니다. 그런 다음 거듭제곱 법칙을 적용하면 지수가 낮아져 방정식이 풀 수 있는 선형 형태로 바뀝니다.지수는 이제 상수와 연수의 곱으로 명확하게 나타납니다.로그 값을 상수로 나누면 인구가 예상되는 최종 인구 크기에 도달하는 데 걸릴 것으로 예상되는 연수를 얻을 수 있습니다.

지수함수 및 로그함수의 방정식: 문제 해결

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&#49373;&#53468;&#54617; &#50672;&#44396;&#50640;&#49436;&#45716; &#50976;&#47532;&#54620; &#51312;&#44148;&#54616;&#50640;&#49436; ...

exponential-equations-with-logarithms-problem-solving

Education

교육

JoVE Core

Math Fundamentals

수학 기초

Unit

단위

Exponential and Logarithmic Functions

지수함수와 로그함수

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Exponential Equations with Logarithms: Problem Solving

로그가 있는 지수 방정식: 문제 해결

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

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