📚 Understanding and Applying Common Logarithms (Base 10) | IB数学：常用对数（以10为底）的理解与应用
Common logarithms, or base-10 logarithms, are one of the most useful tools in mathematics and science. They appear in IB Mathematics SL and HL exams, often combined with exponential equations, functions, and real-world modelling. This guide breaks down what they are, how they work, and how to apply them successfully.
常用对数，即以10为底的对数，是数学和科学中最实用的工具之一。在IB数学SL和HL考试中，它常常与指数方程、函数和实际建模结合出现。本指南将详细讲解它的定义、运算规律以及如何灵活运用。
1. Definition of a Logarithm | 对数的定义
A logarithm answers the question: “To what exponent must a fixed base be raised to produce a given number?” For base 10, we write log₁₀ x or simply log x, which is the power of 10 that equals x. In symbols, y = log₁₀ x means 10ʸ = x.
对数回答的问题是：一个固定的底数需要被提升到多少次方，才能得到给定的数？以10为底时，我们记为log₁₀ x，或简写为log x，它表示等于x所需的10的幂次。用符号表示：y = log₁₀ x 意味着 10ʸ = x。
For example, because 10³ = 1000, we have log₁₀ 1000 = 3. Similarly, 10⁻² = 0.01 gives log₁₀ 0.01 = −2.
例如，由于10³ = 1000，所以log₁₀ 1000 = 3。同理，10⁻² = 0.01，因此log₁₀ 0.01 = −2。
2. History and Notation of Common Logarithms | 常用对数的历史与记号
The common logarithm was introduced by John Napier and later refined by Henry Briggs, who published base-10 logarithm tables in the early 17th century. Before electronic calculators, these tables were essential for astronomical, navigational, and engineering calculations.
常用对数由约翰·纳皮尔提出，后经亨利·布里格斯改进。17世纪初，布里格斯发表了以10为底的对数表。在电子计算器问世之前，这些表格对天文、航海和工程计算至关重要。
In IB examinations, “log x” without a base is assumed to mean base 10. The natural logarithm, written “ln x”, has base e ≈ 2.71828. You must keep the two separate and use the correct key on your calculator.
在IB考试中，未写底数的“log x”默认指以10为底。自然对数写作“ln x”，底数为e ≈ 2.71828。我们必须将二者区分，并在计算器上使用正确的按键。
3. The Three Core Logarithm Laws | 对数三大运算法则
The three core logarithm laws are derived from exponent rules. They apply to all positive bases and positive arguments, and they are the basis for simplifying many algebraic expressions.
三大对数法则由指数法则推导而来。它们适用于任何正底数和正真数，是化简大量代数表达式的基础。
Product rule: The logarithm of a product is the sum of the logarithms of its factors.
积的法则：乘积的对数等于各因子对数之和。
log₁₀ (a × b) = log₁₀ a + log₁₀ b
Quotient rule: The logarithm of a quotient is the difference of the logarithms.
商的法则：商的对数等于被除数的对数减去除数的对数。
log₁₀ (a ÷ b) = log₁₀ a − log₁₀ b
Power rule: The logarithm of a power is the exponent multiplied by the logarithm of the base.
幂的法则：幂的对数等于指数乘以底数的对数。
log₁₀ (aᶜ) = c × log₁₀ a
4. Logarithms as Inverse Functions | 对数与指数互为反函数
The functions y = 10ˣ and y = log₁₀ x are inverses of each other. This means that applying one after the other returns the original input. For example, 10^(log₁₀ 7) = 7 and log₁₀(10⁵) = 5.
函数y = 10ˣ与y = log₁₀ x互为反函数。这意味着连续使用其中一个再使用另一个，会回到原始输入。例如，10^(log₁₀ 7) = 7，而log₁₀(10⁵) = 5。
Graphically, the graph of y = log₁₀ x is a reflection of y = 10ˣ across the line y = x. This inverse relationship is essential when solving exponential equations.
在图像上，y = log₁₀ x的图像是y = 10ˣ关于直线y = x的镜像。这种反函数关系对于解指数方程至关重要。
5. Scientific Notation and Orders of Magnitude | 科学记数法与数量级
Common logarithms give immediate information about the size of a number. If a positive number is written as a × 10ⁿ, where 1 ≤ a < 10, then log₁₀ (a × 10ⁿ) = log₁₀ a + n. Since log₁₀ a is between 0 and 1, the integer part of the logarithm tells us the exponent n, i.e. the order of magnitude.
常用对数能直接给出一个数的大小信息。若一个正数写成a × 10ⁿ，其中1 ≤ a < 10，那么log₁₀ (a × 10ⁿ) = log₁₀ a + n。由于log₁₀ a在0和1之间，所以对数的整数部分就决定了指数n，也就是数量级。
For example, log₁₀ (3.2 × 10⁵) = log₁₀ 3.2 + 5 ≈ 0.505 + 5 = 5.505. Thus the number is of order 10⁵.
例如，log₁₀ (3.2 × 10⁵) = log₁₀ 3.2 + 5 ≈ 0.505 + 5 = 5.505。因此该数的数量级为10⁵。
6. Solving Exponential Equations Involving Base 10 | 求解底数为10的指数方程
When an equation has the unknown in the exponent, taking the common logarithm of both sides is often the fastest way to solve it. For example, to solve 10ˣ = 500, we take log₁₀ of both sides: x = log₁₀ 500 ≈ 2.699.
当方程中的未知数位于指数位置时，对两边取常用对数往往是求解的最快方法。例如，解10ˣ = 500时，两边取log₁₀：x = log₁₀ 500 ≈ 2.699。
For equations like 3ˣ = 20, you cannot use base 10 directly, but you can apply the power rule: x · log₁₀ 3 = log₁₀ 20, so x = log₁₀ 20 ÷ log₁₀ 3 ≈ 2.727.
对于3ˣ = 20这样底数不是10的方程，不能直接使用常用对数，但可以运用幂的法则：x · log₁₀ 3 = log₁₀ 20，因此x = log₁₀ 20 ÷ log₁₀ 3 ≈ 2.727。
7. Graph of y = log₁₀ x | y = log₁₀ x的图像与性质
The graph of y = log₁₀ x has the following key features: its domain is x > 0, its range is all real numbers, and it passes through the points (1, 0) and (10, 1). The y-axis is a vertical asymptote, and the function is strictly increasing.
y = log₁₀ x的图像有以下关键特征：定义域为x > 0，值域为全体实数，经过点(1, 0)和(10, 1)。y轴是垂直渐近线，函数严格递增。
In IB questions, you may be asked to sketch the graph or describe its transformation. Remember that log₁₀ x grows very slowly for large x: log₁₀ 1,000,000 = 6, while x itself is one million.
在IB题目中，可能会要求画图或描述其变换。请记住，对于很大的x，log₁₀ x增长得非常缓慢：log₁₀ 1,000,000 = 6，而x本身已是100万。
8. The Change of Base Formula | 换底公式
The change of base formula lets you compute logarithms with any base using the base-10 logarithm key on your calculator. It states that for any positive base b ≠ 1 and any positive argument x, log_b x = log₁₀ x / log₁₀ b.
换底公式允许我们利用计算器上的常用对数键计算任意底数的对数。它表明：对于任意正底数b ≠ 1和正真数x，log_b x = log₁₀ x / log₁₀ b。
log_b x = log₁₀ x ÷ log₁₀ b
For example, log₂ 8 = log₁₀ 8 ÷ log₁₀ 2 = 0.9031 ÷ 0.3010 = 3. This matches the direct calculation because 2³ = 8.
例如，log₂ 8 = log₁₀ 8 ÷ log₁₀ 2 = 0.9031 ÷ 0.3010 = 3。这与直接计算一致，因为2³ = 8。
9. Common Logarithms vs Natural Logarithms | 常用对数与自然对数的比较
Many IB students wonder why both log and ln exist. The base-10 logarithm is convenient for measurements with powers of 10, while the natural logarithm, base e, is more natural in calculus and continuous growth models. The two are connected by a constant multiplier.
许多IB学生好奇为什么既有log又有ln。以10为底的对数便于处理10的整数次幂，而自然对数以e为底，在微积分和连续增长模型中出现得更自然。二者由一个常数倍相连。
Specifically, ln x = (ln 10) × log₁₀ x. Since ln 10 ≈ 2.3026, we have ln x ≈ 2.3026 · log₁₀ x. On some calculators, you may need to use the change of base formula if only log is available.
具体地，ln x = (ln 10) × log₁₀ x。由于ln 10 ≈ 2.3026，所以ln x ≈ 2.3026 · log₁₀ x。在某些计算器上，如果只有log键，就需要借助换底公式。
10. Application: pH, Earthquakes and Sound | 实际应用：pH值、地震震级与声音分贝
Common logarithms appear in many scientific definitions. The pH of a solution is defined as pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration. A one-unit decrease in pH corresponds to a ten-fold increase in acidity.
常用对数出现在许多科学定义中。溶液的pH值定义为pH = −log₁₀[H⁺]，其中[H⁺]是氢离子浓度。pH每下降1个单位，酸性变为原来的10倍。
The Richter scale uses M = log₁₀(A ÷ A₀), where A is the wave amplitude and A₀ is a reference amplitude. Sound level in decibels is defined as 10 log₁₀(I ÷ I₀). In each case, a multiplicative change in the measured quantity becomes an additive change in the logarithm.
里氏震级使用M = log₁₀(A ÷ A₀)，其中A是地震波振幅，A₀是参考振幅。声音的分贝水平定义为10 log₁₀(I ÷ I₀)。在这类应用中，被测物理量的倍数变化会转化为对数的加法变化。
11. Solving Logarithmic Equations | 解对数方程
When solving an equation involving log₁₀, the first step is often to combine logarithmic terms using the product or quotient rules. For example, log₁₀ x + log₁₀ (x − 3) = log₁₀ 10 can be rewritten as log₁₀ [x(x − 3)] = log₁₀ 10.
解含有log₁₀的方程时，第一步通常是用积或商法则合并对数项。例如，log₁₀ x + log₁₀ (x − 3) = log₁₀ 10可以改写为log₁₀ [x(x − 3)] = log₁₀ 10。
Since the logarithmic function is one-to-one, the equation reduces to x(x − 3) = 10. Solving gives x = 5 or x = −2. However, x = −2 is invalid because the logarithm of a negative number is undefined. Always check that your answers belong to the domain.
因为对数函数是一一对应的，方程可化为x(x − 3) = 10。解得x = 5或x = −2。但x = −2无效，因为负数没有对数。解完必须检查答案是否在定义域内。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
Mistake 1: Treating log
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