Name: logarithms as exponents reversed
Uploaded: 2025-11-26T05:43:44.000Z
Duration: 1 min 14 s
Channel: JoVE

对数函数的底数为不等于1的正数。对数函数产生一个实数输出，该输出表示底数需要被提升到的指数，以获得给定的正输入值。对数函数的一般形式为 y 等于以 a 为底 x 的对数，其中 a 大于零且不等于一。以 e 为底的对数函数称为自然对数函数，记作 ln x。对数函数的图像关于直线 y 等于 x 与指数函数图像对称，且以 y 轴为垂直渐近线，因为对数函数在 x 为零或负数时无定义。对数函数的定义域是所有正实数，值域是所有实数。当底数大于 1 时，对数函数的图像是连续且递增的；当底数介于 0 和 1 之间时，图像为递减的。指数函数在金融领域中用于模拟利息累积的增长过程。

对数函数简介

Introduction to Logarithmic Functions

&#23545;&#25968;&#20989;&#25968;&#26159;&#25351;&#25968;&#20989;&#25968;&#30340;&#21453;&#20989;&#25968;&#65292;&#29992;&#20110;&#27714;&#35299;&#25351;&#25968;&#12290;&#20854;&#19968;&#33324;&#24418;&#24335;&#20026; y = log_a(x)&#65292;&#20854;&#20013; a &gt; 0 &#19988; a &#8800; 1&#12290;&#35813;&#20989;&#25968;&#34920;&#31034;&#24213;&#25968; a &#38656;&#25552;&#21319;&#33267;&#20309;&#31181;&#24130;&#27425;&#26041;&#25165;&#33021;&#24471;&#21040; x&#12290;&#23545;&#25968;&#20989;&#25968;&#20165;&#22312; x &gt; 0 &#26102;&#26377;&#23450;&#20041;&#65292;&#20854;&#20540;&#22495;&#20026;&#25152;&#26377;&#23454;&#25968;&#12290;
&#20174;&#22270;&#20687;&#19978;&#30475;&#65292;&#23545;&#25968;&#20989;&#25968;&#19982;&#25351;&#25968;&#20989;&#25968;&#20114;&#20026;&#20851;&#20110;&#30452;&#32447; y = x &#23545;&#31216;&#12290;&#20989;&#25968; y = log_a(x) &#30340;&#22270;&#20687;&#32463;&#36807;&#28857; (1,0)&#65292;&#24182;&#22312; x = 0 &#22788;&#20855;&#26377;&#22402;&#30452;&#28176;&#36817;&#32447;&#12290;&#35813;&#20989;&#25968;&#22270;&#20687;&#36830;&#32493;&#12289;&#24179;&#28369;&#65292;&#24403; a &gt; 1 &#26102;&#21333;&#35843;&#36882;&#22686;&#65292;&#32780;&#24403; 0 &lt; a &lt; 1 &#26102;&#21333;&#35843;&#36882;&#20943;&#12290;
&#19968;&#31181;&#29305;&#27530;&#30340;&#23545;&#25968;&#20989;&#25968;&#20197;&#24120;&#25968; e&#65288;&#32422;&#20026; 2.718&#65289;&#20316;&#20026;&#24213;&#25968;&#65292;&#31216;&#20026;&#33258;&#28982;&#23545;&#25968;&#65292;&#35760;&#20316; y = ln(x)&#12290;&#33258;&#28982;&#23545;&#25968;&#20989;&#25968; ln(x) &#24191;&#27867;&#24212;&#29992;&#20110;&#28041;&#21450;&#36830;&#32493;&#22686;&#38271;&#30340;&#27169;&#22411;&#20013;&#65292;&#22914;&#20154;&#21475;&#22686;&#38271;&#27169;&#22411;&#21644;&#22797;&#21033;&#35745;&#31639;&#12290;&#19982;&#20854;&#20182;&#23545;&#25968;&#20989;&#25968;&#30456;&#21516;&#65292;ln(x) &#20165;&#22312; x &gt; 0 &#26102;&#23450;&#20041;&#65292;&#19988;&#20026;&#20005;&#26684;&#36882;&#22686;&#20989;&#25968;&#65292;&#20854;&#22270;&#20687;&#21516;&#26679;&#22312; x = 0 &#22788;&#20855;&#26377;&#22402;&#30452;&#28176;&#36817;&#32447;&#12290;

Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Exponential and Logarithmic Functions

指数与对数函数

KEY TERMS AND CONCEPTS

关键术语与概念

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 reflections across the line y = x. The graph of y = log_a(x) passes through (1, 0) and has a vertical asymptote at x = 0. It is continuous and smooth. It increases when a &gt; 1 and decreases when 0 0, is strictly increasing, and has a vertical asymptote at x = 0.

Logarithms as Exponents Reversed

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

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 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

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 logarithms reverse exponents, how their graphs behave, and why ln(x) is useful in growth models like population and compound interest.

Logarithms as Exponents Reversed | 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

Logarithms as Exponents Reversed

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