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

Logarytm wskazuje, ile razy podstawa musi zostać pomnożona przez siebie, aby osiągnąć określoną liczbę. Logarytmy o podstawie 10 są zwykle zapisywane jako "log".Dwa ważne typy praw logarytmicznych to prawo iloczynu i prawo ilorazowe.Prawo produktu stanowi, że logarytm iloczynu jest równy sumie logarytmów.Na przykład logarytm 10 razy 10 jest logarytmem liczby 100, co jest równe 2. Zgodnie z regułą iloczynu jest to równe sumie logarytmów 10 i 10 — z których oba są równe 1, ponieważ logarytm liczby o własnej podstawie wynosi 1, co daje w sumie 2.Prawo ilorazu mówi, że logarytm ilorazu jest równy różnicy logarytmów.Na przykład logarytm liczby 10 podzielony przez 10 jest logarytmem liczby 1, który jest równy 0, ponieważ logarytm liczby 1 o dowolnej podstawie wynosi 0. Reguła ilorazu pokazuje, że jest to równa się różnicy między logarytmami 10 i 10 — co daje 0.Na przykład skala pH wykorzystuje logarytmy do pomiaru kwasowości na podstawie stężenia jonów wodorowych. Jeśli stężenie wynosi 2 razy 10 do potęgi minus 7 mol/l, pH jest ujemnym logarytmem tego. Ustawa o produkcie upraszcza tę kwestię do punktu 6.7.

prawa logarytmów

Logarytmy s&#261; podstawowymi operacjami matematycznymi, stanowi&#261;cymi odwrotno&#347;&#263; pot&#281;gowania. Pozwalaj&#261; wyrazi&#263;, ile razy nale&#380;y podnie&#347;&#263; podstaw&#281;, aby otrzyma&#263; dan&#261; liczb&#281;. Dla podstawy 10, cz&#281;sto nazywanej logarytmem dziesi&#281;tnym, stosuje si&#281; zapis log. Zatem, je&#347;li 10^n = x, to log (x) = n. Ta zale&#380;no&#347;&#263; czyni logarytmy szczeg&#243;lnie u&#380;ytecznymi przy upraszczaniu z&#322;o&#380;onych oblicze&#324; obejmuj&#261;cych mno&#380;enie, dzielenie i pot&#281;gowanie.Wyra&#380;enia logarytmiczne podlegaj&#261; podstawowym prawom, niezb&#281;dnym m.in. przy rozwi&#261;zywaniu nier&#243;wno&#347;ci nieliniowych. Prawo iloczynu logarytm&#243;w g&#322;osi, &#380;e logarytm iloczynu jest r&#243;wny sumie logarytm&#243;w:<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>Na przyk&#322;ad log (100) mo&#380;na zapisa&#263; jako log (10&#215;10). Zastosowanie prawa iloczynu daje log (10)+log (10), co daje 2, zgodnie z faktem, &#380;e 10^2 = 100. Regu&#322;a ta jest szczeg&#243;lnie u&#380;yteczna do rozbijania du&#380;ych warto&#347;ci na &#322;atwiejsze sk&#322;adniki, usprawniaj&#261;c obliczenia bez bezpo&#347;redniego mno&#380;enia.Prawo ilorazu logarytm&#243;w wyra&#380;a logarytm ilorazu jako r&#243;&#380;nic&#281; logarytm&#243;w:<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>Na przyk&#322;ad log (1) mo&#380;na zapisa&#263; jako log (10/10). Zgodnie z prawem ilorazu sprowadza si&#281; to do log (10)&#8722;log (10)=1&#8722;1=0. Wynik ten jest zgodny z w&#322;asno&#347;ci&#261;, &#380;e logarytm z 1 przy dowolnej podstawie jest r&#243;wny zeru, poniewa&#380; ka&#380;da liczba podniesiona do pot&#281;gi zerowej daje jeden.Prawa logarytmiczne maj&#261; praktyczne zastosowania m.in. w akustyce i sejsmologii. Poziom d&#378;wi&#281;ku mierzy si&#281; w decybelach, a magnitudy trz&#281;sie&#324; ziemi &#8212; w skali Richtera, obie te skale s&#261; logarytmiczne. W tych systemach ka&#380;dy wzrost o jedn&#261; jednostk&#281; oznacza znaczny przyrost intensywno&#347;ci lub energii, odzwierciedlaj&#261;c nieliniowy charakter zjawisk. Nale&#380;y tak&#380;e upewni&#263; si&#281;, &#380;e argumenty wszystkich wyra&#380;e&#324; logarytmicznych s&#261; dodatnie, gdy&#380; logarytmy s&#261; niezdefiniowane dla warto&#347;ci niedodatnich. Ten warunek dotycz&#261;cy dziedziny jest kluczowy przy wyznaczaniu poprawnych rozwi&#261;za&#324;.

Prawa logarytmów

Logarytmy s&#261; podstawowymi operacjami matematycznymi, stanowi&#261;cymi odwrotno&#347;&#263; pot&#281;gowania. Pozwalaj&#261; wyrazi&#263;, ile razy ...

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JoVE Core

Math Fundamentals

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Jednostka

Exponential and Logarithmic Functions

Funkcje wykładnicze i logarytmiczne

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

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

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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

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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