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

Geniş bir kunduz habitatına sahip ormanlık bir bölgede, bir araştırmacı kunduz popülasyonunun zaman içinde nasıl büyüdüğünü dikkatle takip ediyor.Amaç, popülasyonun belirli bir büyüklüğe ulaşması için gereken yıl sayısını belirlemektir.Nüfus, zaman içinde tekrarlanan büyümeye dayalı üstel bir model izler. İlk popülasyonun 10 ile çarpımının, büyüme hızı çarpı yıl sayısına eşittir. Büyüme hızı, nüfusun her yıl ne kadar hızlı arttığını gösterir.Hesaplamaya başlamak için araştırmacı, hedef popülasyon değerini denklemde yerine koyar.Her iki tarafı da ilk nüfusa bölmek, nüfusun arttığı faktörü verir. Denklem daha sonra yeniden düzenlenir, böylece bir üsse yükseltilmiş on bu faktöre eşit olur.Logaritmalar ve üsler ters işlemler olduğundan, her iki tarafın logaritmasını almak değişkeni izole eder. Daha sonra kuvvet yasasını uygulamak üssü aşağı indirerek denklemi çözülebilir doğrusal bir forma dönüştürür.Üs şimdi açıkça sabitin ve yıl sayısının bir ürünü olarak görünüyor.Logaritmik değerin sabite bölünmesi, popülasyonun beklenen nihai popülasyon büyüklüğüne ulaşması için gereken tahmini yıl sayısını verir.

Üstel ve logaritmik denklemler: problem Çözme

Ekolojik &#231;al&#305;&#351;malarda, uygun ko&#351;ullar alt&#305;nda pop&#252;lasyonlar&#305;n zaman i&#231;inde nas&#305;l b&#252;y&#252;d&#252;&#287;&#252;n&#252; tahmin etmek i&#231;in s&#305;kl&#305;kla &#252;stel modeller kullan&#305;l&#305;r. Bu modeller, b&#252;y&#252;me oran&#305;n&#305;n mevcut pop&#252;lasyonla orant&#305;l&#305; oldu&#287;unu varsayar; bu da s&#252;rekli ve bile&#351;ik art&#305;&#351;lara yol a&#231;ar.
Model, pop&#252;lasyonu zaman&#305;n bir fonksiyonu olarak ifade eder ve ba&#351;lang&#305;&#231; pop&#252;lasyonunu, b&#252;y&#252;me oran&#305; ile zaman&#305; i&#231;eren bir &#252;sse y&#252;kseltilmi&#351; bir b&#252;y&#252;me fakt&#246;r&#252;yle &#231;arpar. Bir pop&#252;lasyonun belirli bir b&#252;y&#252;kl&#252;&#287;e ula&#351;mas&#305;n&#305;n ne kadar s&#252;rd&#252;&#287;&#252;n&#252; tahmin etmek i&#231;in ara&#351;t&#305;rmac&#305;lar, hedef pop&#252;lasyon b&#252;y&#252;kl&#252;&#287;&#252;n&#252; modele yerine koyar ve ba&#351;lang&#305;&#231; de&#287;erine b&#246;lerler. Bu i&#351;lem, pop&#252;lasyonun ka&#231; kat&#305;na &#231;&#305;kt&#305;&#287;&#305;n&#305; g&#246;steren bir b&#252;y&#252;me fakt&#246;r&#252; verir.
Y&#305;l say&#305;s&#305; b&#252;y&#252;me ifadesinin i&#231;inde yer ald&#305;&#287;&#305;ndan, bunu belirlemek &#252;stel b&#252;y&#252;me s&#252;recini tersine &#231;evirmeyi gerektirir. Bu, zaman&#305; ba&#351;lang&#305;&#231; ve son pop&#252;lasyon b&#252;y&#252;kl&#252;kleri ile b&#252;y&#252;me oran&#305; gibi bilinen nicelikler cinsinden ifade etmeye yard&#305;mc&#305; olan logaritmik ak&#305;l y&#252;r&#252;tme kullan&#305;larak yap&#305;l&#305;r. Logaritmik ak&#305;l y&#252;r&#252;tme yoluyla bilginin yeniden yap&#305;land&#305;r&#305;lmas&#305;yla, zaman do&#287;rudan hesaplanabilir bir b&#252;y&#252;kl&#252;&#287;e d&#246;n&#252;&#351;&#252;r ve pop&#252;lasyonun tutarl&#305; b&#252;y&#252;me ko&#351;ullar&#305;nda hedef b&#252;y&#252;kl&#252;&#287;e ula&#351;mas&#305;n&#305;n ne kadar s&#252;rece&#287;i ortaya konur.
Bu durum, logaritmalar&#305;n &#252;stel denklemleri &#231;&#246;zmek i&#231;in nas&#305;l kullan&#305;ld&#305;&#287;&#305;n&#305; vurgular ve bir pop&#252;lasyonun istenen b&#252;y&#252;kl&#252;&#287;e ula&#351;mas&#305; i&#231;in gereken s&#252;renin tahmin edilmesini m&#252;mk&#252;n k&#305;lar. Pop&#252;lasyon modellemesi ve kaynak y&#246;netiminde temel bir ara&#231;t&#305;r.

Üstel ve Logaritmik Denklemler: Problem Çözme

Ekolojik &#231;al&#305;&#351;malarda, uygun ko&#351;ullar alt&#305;nda pop&#252;lasyonlar&#305;n zaman i&#231;inde nas&#305;l ...

exponential-equations-with-logarithms-problem-solving

Education

JoVE Core

Math Fundamentals

Matematik Temelleri

Unit

Birim

Exponential and Logarithmic Functions

Üstel ve Logaritmik Fonksiyonlar

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 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...

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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...

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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...

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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...

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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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