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

Un logarithme indique combien de fois une base doit être multipliée par elle-même pour atteindre un nombre spécifique. Les logarithmes en base 10 s’écrivent généralement 'log'.Deux types importants de lois logarithmiques sont la loi du produit et la loi du quotient.La loi du produit stipule que le logarithme d’un produit est égal à la somme des logarithmes.Par exemple, le logarithme de 10 fois 10 est le logarithme de 100, ce qui est égal à 2. Selon la règle du produit, cela équivaut à la somme des logarithmes de 10 et 10 - qui sont tous deux égaux à 1, car le logarithme d’un nombre à sa propre base est 1, ce qui donne un total de 2.La loi du quotient stipule que le logarithme d’un quotient est égal à la différence des logarithmes.Par exemple, le logarithme de 10 divisé par 10 est le logarithme de 1, qui est égal à 0 puisque le logarithme de 1 à une base quelconque est 0. La règle du quotient montre que cela est égal à la différence entre les logarithmes de 10 et 10, ce qui donne 0.L’échelle de pH, par exemple, utilise des logarithmes pour mesurer l’acidité à partir de la concentration en ions hydrogène. Si la concentration est de 2 fois 10 à la puissance moins 7 mol/L, le pH est le logarithme négatif de celle-ci. La loi sur les produits simplifie cela à 6.7.

lois des logarithmes

Laws of Logarithms I

Les logarithmes sont des op&#233;rations math&#233;matiques fondamentales, inverses de l'exponentiation. Ils permettent d'exprimer combien de fois il faut &#233;lever une base pour obtenir un nombre donn&#233;. En base 10, souvent appel&#233;e logarithme d&#233;cimal, la notation est simplement log. Ainsi, si 10^n = x, alors log (x) = n. Cette relation rend les logarithmes particuli&#232;rement utiles pour simplifier des calculs complexes impliquant la multiplication, la division et l'exponentiation.Les expressions logarithmiques sont r&#233;gies par des lois fondamentales essentielles &#224; la r&#233;solution d'in&#233;galit&#233;s non lin&#233;aires. La loi du produit stipule que le logarithme d'un produit est &#233;gal &#224; la somme des logarithmes de ce produit.<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>Par exemple, log (100) peut s&#8217;exprimer sous la forme : log (10&#215;10). L&#8217;application de la loi du produit donne log (10) + log (10) = 2, en accord avec le fait que 10^2 = 100. Cette r&#232;gle est particuli&#232;rement utile pour d&#233;composer de grandes valeurs en composantes g&#233;rables, simplifiant ainsi les calculs sans multiplication directe.La loi du quotient exprime le logarithme d'un quotient comme la diff&#233;rence entre les logarithmes&#160;:<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>Par exemple, log (1) peut s&#8217;&#233;crire log (10/10). Selon la loi du quotient, cela se simplifie en log (10) &#8211; log (10) = 1&#8722;1 = 0. Ce r&#233;sultat est conforme &#224; la propri&#233;t&#233; selon laquelle le logarithme de 1, quelle que soit sa base, est toujours nul, puisque tout nombre &#233;lev&#233; &#224; la puissance z&#233;ro est &#233;gal &#224; 1.Les lois logarithmiques ont une importance concr&#232;te dans des domaines tels que l&#8217;acoustique et la sismologie. Les niveaux sonores sont mesur&#233;s en d&#233;cibels et les magnitudes sismiques sur l&#8217;&#233;chelle de Richter, toutes deux recourent &#224; des &#233;chelles logarithmiques. Dans ces syst&#232;mes, chaque augmentation d&#8217;une unit&#233; repr&#233;sente une hausse substantielle de l&#8217;intensit&#233; ou de l&#8217;&#233;nergie, refl&#233;tant le comportement non lin&#233;aire des ph&#233;nom&#232;nes. Il est &#233;galement essentiel de s&#8217;assurer que les arguments de toutes les expressions logarithmiques restent strictement positifs, car les logarithmes sont non d&#233;finis pour les valeurs non positives. Cette condition de d&#233;finition est essentielle pour d&#233;terminer des solutions valides.

Lois des logarithmes

Les logarithmes sont des op&#233;rations math&#233;matiques fondamentales, inverses de l'exponentiation. Ils permettent d'exprimer combien de fois il faut ...

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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 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

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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 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...

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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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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.

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Logarithm Rules for Product and Quotient

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