Name: logarithm rules for product and quotient
Uploaded: 2025-11-26T05:43:42.000Z
Duration: 1 min 30 s
Channel: JoVE

Um logaritmo indica quantas vezes uma base deve ser multiplicada por ela mesma para atingir um número específico. Os logaritmos de base 10 são geralmente escritos como 'log'.Dois tipos importantes de leis logarítmicas são a lei do produto e a lei do quociente.A lei dos produtos afirma que o logaritmo de um produto é igual à soma dos logaritmos.Por exemplo, o logaritmo de 10 vezes 10 é o logaritmo de 100, que é igual a 2. De acordo com a regra do produto, isso é igual à soma dos logaritmos de 10 e 10 - ambos são 1, porque o logaritmo de um número em sua própria base é 1, dando um total de 2.A lei do quociente afirma que o logaritmo de um quociente é igual à diferença dos logaritmos.Por exemplo, o logaritmo de 10 dividido por 10 é o logaritmo de 1, que é igual a 0, pois o logaritmo de 1 para qualquer base é 0. A regra do quociente mostra que isso é igual à diferença entre os logaritmos de 10 e 10 - dando 0.A escala de pH, por exemplo, usa logaritmos para medir a acidez da concentração de íons de hidrogênio. Se a concentração for 2 vezes 10 elevado a menos 7 mol/L, o pH é o logaritmo negativo disso. A lei do produto simplifica isso para 6.7.

leis dos logaritmos

Logaritmos s&#227;o opera&#231;&#245;es matem&#225;ticas fundamentais que atuam como o inverso da exponencia&#231;&#227;o. Eles fornecem um meio de expressar quantas vezes uma base deve ser elevada para obter um determinado n&#250;mero. Para a base 10, frequentemente chamada de logaritmo comum, a nota&#231;&#227;o &#233; simplesmente log. Assim, se 10^n = x, ent&#227;o log (x) = n. Essa rela&#231;&#227;o torna os logaritmos especialmente &#250;teis na simplifica&#231;&#227;o de c&#225;lculos complexos que envolvem multiplica&#231;&#227;o, divis&#227;o e exponencia&#231;&#227;o.Express&#245;es logar&#237;tmicas s&#227;o regidas por leis fundamentais, essenciais na resolu&#231;&#227;o de desigualdades n&#227;o lineares. A lei do produto afirma que o logaritmo de um produto &#233; igual &#224; soma dos logaritmos:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40063/40063_Equation_1_2__editor_40063.svg' alt='40063_Equation_1_2_.svg' /></figure>Por exemplo, log (100) pode ser expresso como log (10 &#215; 10). Aplicando a lei do produto obt&#233;m-se que log (10) + log (10) = 2, o que concorda com 10^2 = 100. Essa regra &#233; particularmente &#250;til para decompor valores grandes em componentes manej&#225;veis, simplificando c&#225;lculos sem realizar a multiplica&#231;&#227;o direta.A lei do quociente expressa o logaritmo de um quociente como a diferen&#231;a entre os logaritmos:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40063/40063_Equation_2_2__editor_40063.svg' alt='40063_Equation_2_2_.svg' /></figure>Por exemplo, log (1) pode ser escrito como log (10/10). De acordo com a lei do quociente, isso se simplifica para log (10) &#8211; log (10) = 1 &#8722; 1 = 0. Esse resultado est&#225; em conformidade com a propriedade de que o logaritmo de 1, em qualquer base, &#233; sempre zero, j&#225; que qualquer n&#250;mero elevado a zero &#233; igual a um.As leis logar&#237;tmicas t&#234;m relev&#226;ncia pr&#225;tica em campos como a ac&#250;stica e a sismologia. Os n&#237;veis sonoros s&#227;o medidos em decib&#233;is e as magnitudes dos terremotos na escala Richter, ambas utilizando escalas logar&#237;tmicas. Nesses sistemas, cada aumento unit&#225;rio representa um acr&#233;scimo substancial na intensidade ou na energia, refletindo o comportamento n&#227;o linear dos fen&#244;menos. &#201; igualmente essencial garantir que os argumentos de todas as express&#245;es logar&#237;tmicas permane&#231;am positivos, uma vez que os logaritmos s&#227;o indefinidos para valores n&#227;o positivos. Essa condi&#231;&#227;o de dom&#237;nio &#233; crucial para determinar solu&#231;&#245;es v&#225;lidas.

Leis dos Logaritmos

Logaritmos s&#227;o opera&#231;&#245;es matem&#225;ticas fundamentais que atuam como o inverso da exponencia&#231;&#227;o. Eles fornecem um meio de ...

Education

JoVE Core

Math Fundamentals

Fundamentos da Matemática

Unit

Unidade

Exponential and Logarithmic Functions

Funções Exponenciais e Logarítmicas

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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 log(10 × 10). Using the product law gives log(10) + log(10), which equals 2. That matches the fact that 10^2 = 100.

The quotient law works the same way for division. It says that the logarithm of a quotient equals the difference of the logarithms. For example, log(1) can be written as log(10/10). The quotient law gives log(10) - log(10) = 1 - 1 = 0. This also fits the rule that the logarithm of 1 is always zero, because any number raised to the zero power equals one.

Logarithmic laws are important in real-world settings such as acoustics and seismology. Sound levels use decibels, and earthquake strength uses the Richter scale. Both scales are logarithmic, so each unit increase means a large rise in intensity or energy. When solving logarithmic expressions, it is also essential to keep the argument positive, because logarithms are undefined for non-positive values.

Logarithm Rules for Product and Quotient

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...

laws of logarithms i

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.

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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...

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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 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 product and quotient laws simplify logarithms, why log(1)=0, and how logarithmic scales appear in decibels and earthquake measurements.

Logarithm Rules for Product and Quotient | 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

Logarithm Rules for Product and Quotient

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