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

対数は、特定の数に達するために底を何回乗算する必要があるかを示します。10進法の対数は通常「log」と表記される。対数法則の2つの重要なタイプは、積の法則と商の法則です。積の法則では、積の対数は対数の合計に等しいと規定されています。たとえば、10 に 10 を掛けた対数は 100 の対数であり、2 に等しくなります。積則によれば、これは10と10の対数の合計に等しくなります—数値の底に対する対数は1であり、合計は2になるため、どちらも1です。商の法則は、商の対数が対数の差に等しいと述べています。たとえば、10の対数を10で割ったものは1の対数であり、任意の底に対する1の対数は0であるため、0に等しくなります。商則は、これが10と10の対数の差に等しいことを示し、0になります。たとえば、pH スケールは対数を使用して水素イオン濃度から酸性度を測定します。濃度が負の7 mol / Lの10乗の2倍の場合、pHはその負の対数です。製品法則はこれを6.7に簡略化しています。

対数の法則

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log (10&#215;10) &#12392;&#34920;&#12377;&#12371;&#12392;&#12364;&#12391;&#12365;&#12414;&#12377;&#12290;&#31309;&#12398;&#27861;&#21063;&#12434;&#36969;&#29992;&#12377;&#12427;&#12392;&#12289;log (10) + log (10) = 2 &#12392;&#12394;&#12426;&#12289;&#12371;&#12428;&#12399; 10^2 = 100 &#12392;&#12356;&#12358;&#20107;&#23455;&#12392;&#19968;&#33268;&#12375;&#12414;&#12377;&#12290;&#12371;&#12398;&#27861;&#21063;&#12399;&#12289;&#22823;&#12365;&#12394;&#20516;&#12434;&#25201;&#12356;&#12420;&#12377;&#12356;&#35201;&#32032;&#12395;&#20998;&#12369;&#12289;&#30452;&#25509;&#20055;&#31639;&#12377;&#12427;&#12371;&#12392;&#12394;&#12367;&#35336;&#31639;&#12434;&#21177;&#29575;&#21270;&#12377;&#12427;&#12398;&#12395;&#29305;&#12395;&#24441;&#31435;&#12385;&#12414;&#12377;&#12290;&#21830;&#12398;&#27861;&#21063;&#12399;&#12289;&#12300;&#21830;&#12398;&#23550;&#25968;&#12399;&#23550;&#25968;&#12398;&#24046;&#12395;&#31561;&#12375;&#12356;&#12301;&#12392;&#31034;&#12375;&#12390;&#12356;&#12414;&#12377;&#12290;<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:48.25%;'><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>&#20363;&#12360;&#12400;&#12289;log (1) &#12399; 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Education

JoVE Core

Math Fundamentals

数学の基礎

Unit

ユニット

Exponential and Logarithmic Functions

指数関数と対数関数

KEY TERMS AND CONCEPTS

主要用語と概念

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...

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 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

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...

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 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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