📚 IB Math: Logarithm Rules Summary | IB数学：对数运算法则归纳
Logarithms are a fundamental tool in IB Mathematics, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI). Mastering the rules of logarithms is essential for solving exponential equations, modelling growth and decay, and simplifying complex expressions. This guide summarises the core logarithm laws, provides examples, and highlights common pitfalls to help you prepare for exams.
对数是IB数学中的基础工具，无论在分析与方法（AA）还是应用与解释（AI）课程中都会频繁出现。掌握对数运算法则是解指数方程、建模增长与衰减以及化简复杂表达式的前提。本指南系统归纳了对数核心法则，提供示例，并指出常见易错点，助你备考。
1. Definition of Logarithms | 对数的定义
A logarithm answers the question: “To what exponent must a given base be raised to produce a certain number?” If \(a^x = b\), then \(\log_a b = x\), where \(a > 0\), \(a \neq 1\), and \(b > 0\).
对数回答的问题是：“底数需要被提升到多少次方，才能得到某个数？”若 \(a^x = b\)，则 \(\log_a b = x\)，其中 \(a > 0\)，\(a \neq 1\)，且 \(b > 0\)。
For example, since \(2^3 = 8\), we have \(\log_2 8 = 3\). The base 2 is raised to the power 3 to give 8. This inverse relationship is central to all logarithm rules.
例如，因为 \(2^3 = 8\)，所以 \(\log_2 8 = 3\)。底数2被提升到3次方得到8。这种反函数关系是所有对数法则的核心。
2. Relationship with Exponents | 对数与指数的关系
Logarithms and exponentials are inverse operations. For any base \(a > 0\), \(a \neq 1\):
对数与指数是互逆运算。对任意底数 \(a > 0\)，\(a \neq 1\)：
\(\log_a (a^x) = x\) and \(a^{\log_a x} = x\)
These two identities are often called the “cancellation laws”. They are extremely useful when simplifying expressions that mix logs and exponentials. For instance, \(\log_5 (5^{2x+1}) = 2x+1\) and \(3^{\log_3 7} = 7\).
这两个恒等式常被称为“消去法则”。在化简同时含有对数与指数的表达式时非常有用。例如，\(\log_5 (5^{2x+1}) = 2x+1\)，且 \(3^{\log_3 7} = 7\)。
In IB exams, always check the domain: the argument of a logarithm must be positive, and the base must be positive and not equal to 1. These restrictions affect the validity of any logarithmic manipulation.
在IB考试中，始终要检查定义域：对数的真数必须为正，底数必须为正且不等于1。这些限制会影响所有对数运算的有效性。
3. Product Rule | 乘积法则
The logarithm of a product equals the sum of the logarithms of the factors. For \(M > 0\), \(N > 0\), and base \(a > 0, a \neq 1\):
乘积的对数等于各因子对数之和。对于 \(M > 0\)，\(N > 0\)，底数 \(a > 0, a \neq 1\)：
\(\log_a (MN) = \log_a M + \log_a N\)
Example: \(\log_2 32 = \log_2 (4 \times 8) = \log_2 4 + \log_2 8 = 2 + 3 = 5\). This rule is especially useful when the argument is a product of integers or algebraic expressions.
例：\(\log_2 32 = \log_2 (4 \times 8) = \log_2 4 + \log_2 8 = 2 + 3 = 5\)。当真数是整数或代数表达式的乘积时，这一法则尤其有用。
Be careful not to over-apply the rule: \(\log_a (M + N)\) is NOT equal to \(\log_a M + \log_a N\). The product rule only applies to multiplication inside the logarithm.
注意不要滥用该法则：\(\log_a (M + N)\) 不等于 \(\log_a M + \log_a N\)。乘积法则只适用于对数内部的乘法。
4. Quotient Rule | 商法则
The logarithm of a quotient equals the difference of the logarithms of the numerator and denominator. For \(M > 0\), \(N > 0\):
商的对数等于分子对数与分母对数之差。对于 \(M > 0\)，\(N > 0\)：
\(\log_a \left(\frac{M}{N}\right) = \log_a M – \log_a N\)
Example: \(\log_3 \left(\frac{81}{9}\right) = \log_3 81 – \log_3 9 = 4 – 2 = 2\). This rule is efficient for simplifying rational expressions inside a logarithm.
例：\(\log_3 \left(\frac{81}{9}\right) = \log_3 81 – \log_3 9 = 4 – 2 = 2\)。该法则可用于简化对数内部的分数表达式。
Note that \(\log_a M – \log_a N\) is not the same as \(\frac{\log_a M}{\log_a N}\). Subtraction of logs corresponds to division of the original arguments, not division of the logs themselves.
注意：\(\log_a M – \log_a N\) 不等于 \(\frac{\log_a M}{\log_a N}\)。对数的差对应原真数的除法，而不是对数本身的除法。
5. Power Rule | 幂法则
The logarithm of a number raised to a power equals the power multiplied by the logarithm of the number. For \(M > 0\) and any real exponent \(p\):
一个数的幂的对数等于幂指数乘以该数的对数。对于 \(M > 0\) 和任意实数指数 \(p\)：
\(\log_a (M^p) = p \cdot \log_a M\)
Example: \(\log_5 (125^{2}) = 2 \log_5 125 = 2 \times 3 = 6\). This rule also allows us to bring variable exponents down, which is the key step when solving exponential equations.
例：\(\log_5 (125^{2}) = 2 \log_5 125 = 2 \times 3 = 6\)。该法则还能将变量指数移到前面，这是解指数方程的关键步骤。
The power rule works in both directions: you can move a coefficient up as an exponent, such as \(3 \log_2 x = \log_2 (x^3)\). However, this is only valid when \(x > 0\). For even powers, the domain may require absolute values: \(\log_a (x^2) = 2 \log_a |x|\) when \(x\) can be any non-zero real number.
幂法则可以双向使用：可以将系数移到上方作为指数，如 \(3 \log_2 x = \log_2 (x^3)\)。但仅当 \(x > 0\) 时成立。对于偶次幂，定义域可能需要绝对值：当 \(x\) 可以为任意非零实数时，\(\log_a (x^2) = 2 \log_a |x|\)。
6. Change of Base Formula | 换底公式
Sometimes it is necessary to convert a logarithm from one base to another, especially when using a calculator that only handles base 10 or base \(e\). The change of base formula states that for any positive bases \(a, b\) with \(a \neq 1, b \neq 1\):
有时需要将对数从一个底数转换为另一个底数，尤其是当计算器只支持以10或 \(e\) 为底时。换底公式指出，对于任意正底数 \(a, b\) 且 \(a \neq 1, b \neq 1\)：
\(\log_a M = \frac{\log_b M}{\log_b a}\)
For example, \(\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{0.3010} \approx 3.3219\). In IB papers, it is common to express the answer in terms of natural logarithms: \(\log_2 10 = \frac{\ln 10}{\ln 2}\).
例如，\(\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2} = \frac{1}{0.3010} \approx 3.3219\)。在IB试卷中，常要求用自然对数表示答案：\(\log_2 10 = \frac{\ln 10}{\ln 2}\)。
A useful consequence is that \(\log_a b \cdot \log_b a = 1\), which can simplify products of logs with different bases.
一个有用的推论是 \(\log_a b \cdot \log_b a = 1\)，可用于化简不同底数对数的乘积。
7. Common and Natural Logarithms | 常用对数与自然对数
The common logarithm has base 10 and is written as \(\log_{10} x\) or simply \(\log x\). The natural logarithm has base \(e\) and is written as \(\log_e x\) or \(\ln x\). Both are accepted in IB examinations, but you must follow the notation specified on your calculator and in the question.
常用对数以10为底，记为 \(\log_{10} x\) 或简写为 \(\log x\)。自然对数以 \(e\) 为底，记为 \(\log_e x\) 或 \(\ln x\)。两者在IB考试中均可使用，但必须遵循题目和计算器指定的记号。
Natural logarithms have special importance in calculus because the derivative of \(\ln x\) is \(1/x\). In IB AA, you are expected to know that \(\ln e = 1\) and \(e^{\ln x} = x\).
自然对数在微积分中尤为重要，因为 \(\ln x\) 的导数为 \(1/x\)。在IB AA课程中，你需要知道 \(\ln e = 1\) 和 \(e^{\ln x} = x\)。
When solving equations, you can take \(\ln\) or \(\log\) on both sides. For example, to solve \(2^x = 5\), take natural logs: \(\ln(2^x) = \ln 5\), so \(x \ln 2 = \ln 5\), hence \(x = \frac{\ln 5}{\ln 2}\).
解方程时，可以对两边取 \(\ln\) 或 \(\log\)。例如，解 \(2^x = 5\)，两边取自然对数：\(\ln(2^x) = \ln 5\)，所以 \(x \ln 2 = \ln 5\)，因此 \(x = \frac{\ln 5}{\ln 2}\)。
8. Key Identities | 关键恒等式
The following identities are direct consequences of the definition and the rules above. They frequently appear in exam problems without explicit mention.
以下恒等式是对数定义和上述法则的直接推论，在考试题目中常常被隐式使用。
\(\log_a 1 = 0\) \(\log_a a = 1\) \(\log_a \frac{1}{a} = -1\)
Also, \(\log_a (M^k) = k \log_a M\) and \(\log_a \sqrt[n]{M} = \frac{1}{n} \log_a M\). For example, \(\log_7 \sqrt[3]{7} = \frac{1}{3}\). Remember that \(\log_a a^x = x\) works for any real \(x\).
另外，\(\log_a (M^k) = k \log_a M\)，且 \(\log_a \sqrt[n]{M} = \frac{1}{n} \log_a M\)。例如，\(\log_7 \sqrt[3]{7} = \frac{1}{3}\)。记住 \(\log_a a^x = x\) 对任意实数 \(x\) 都成立。
A combined identity worth memorising is \(\log_a M \times \log_N a = \log_N M\). This follows from the change of base formula and is useful when bases cancel.
一个值得记忆的复合恒等式是 \(\log_a M \times \log_N a = \log_N M\)。它由换底公式推出，在底数相消时非常有用。
9. Solving Logarithmic and Exponential Equations | 解对数与指数方程
To solve equations involving exponentials, apply logarithms to both sides and use the power rule. For example, solve \(3^{2x} = 4^{x-1}\): taking logs gives \(2x \ln 3 = (x-1) \ln 4\), then expand and solve for \(x\).
解含指数的方程时，两边取对数并利用幂法则。例如，解 \(3^{2x} = 4^{x-1}\)：两边取对数得 \(2x \ln 3 = (x-1) \ln 4\)，然后展开并解出 \(x\)。
For logarithmic equations, use the rules to combine logs into a single log, then exponentiate. For instance, \(\log_2 x + \log_2 (x+2) = 3\) becomes \(\log_2 [x(x+2)] = 3\), so \(x(x+2) = 2^3 = 8\), yielding \(x = 2\) or \(x = -4\). Always reject solutions that make the original argument non-positive; here \(x = -4\) is invalid because \(\log_2(-4)\) is undefined.
解对数方程时，先利用法则将对数合并成一个对数，再取指数。例如，\(\log_2 x + \log_2 (x+2) = 3\) 可化简为 \(\log_2 [x(x+2)] = 3\)，因此 \(x(x+2) = 2^3 = 8\)，得到 \(x = 2\) 或 \(x = -4\)。必须舍去使原真数非正的解；这里 \(x = -4\) 不合法，因为 \(\log_2(-4)\) 无定义。
When the equation has logarithms on both sides, \(\log_a f(x) = \log_a g(x)\), you can equate the arguments: \(f(x) = g(x)\), provided both are in the domain. Check the solution at the end.
当方程两边都有对数时，如 \(\log_a f(x) = \log_a g(x)\)，可令真数相等：\(f(x) = g(x)\)，前提是两者均在定义域内。最终必须验根。
10. Common Errors and Pitfalls | 常见错误与陷阱
Below are frequent mistakes made by IB students when working with logarithms, along with the correct interpretation.
以下是IB学生在处理对数时常犯的错误及其正确理解。
-
Error: \(\log_a (M + N) = \log_a M + \log_a N\)
Correct: \(\log_a (M + N)\) cannot be simplified further. -
错误： \(\log_a (M + N) = \log_a M + \log_a N\)
正确： \(\log_a (M + N)\) 无法进一步化简。 -
Error: \(\frac{\log_a M}{\log_a N} = \log_a M – \log_a N\)
Correct: \(\log_a M – \log_a N = \log_a \frac{M}{N}\), not a ratio of logs. -
错误： \(\frac{\log_a M}{\log_a N} = \log_a M – \log_a N\)
正确： \(\log_a M – \log_a N = \log_a \frac{M}{N}\)，而非对数的比值。 -
Error: Forgetting the domain: \(\log_a x^2 = 2 \log_a x\) when \(x < 0\).
Correct: Use \(\log_a x^2 = 2 \log_a |x|\), or restrict \(x > 0\). -
错误： 忽略定义域：当 \(x < 0\) 时仍写 \(\log_a x^2 = 2 \log_a x\)。
正确： 应写 \(\log_a x^2 = 2 \log_a |x|\)，或限制 \(x > 0\)。 -
Error: Assuming base 10 when no base is written: \(\log 100\) is base 10, but \(\ln 100\) is base \(e\).
-
错误： 默认未写底数时是以10为底：\(\log 100\) 确实是底数10，但 \(\ln 100\) 是以 \(e\) 为底。
-
Error: Changing the base incorrectly: \(\log_a b = \frac{\log b}{\log a}\), not \(\frac{\log a}{\log b}\).
-
错误： 换底公式写错：\(\log_a b = \frac{\log b}{\log a}\)，而不是 \(\frac{\log a}{\log b}\)。
11. Applications in IB Problems | IB题目中的应用
Logarithm rules are essential in solving problems involving exponential growth and decay, such as radioactive decay or compound interest. For example, if a population doubles every 3 years, the formula \(P = P_0 2^{t/3}\) can be solved for \(t\) using logs: \(t = 3 \log_2 (P/P_0)\).
对数法则在解指数增长与衰减问题（如放射性衰变或复利）中至关重要。例如，某人口每3年翻倍，公式 \(P = P_0 2^{t/3}\) 可通过取对数解出 \(t\)：\(t = 3 \log_2 (P/P_0)\)。
In IB AA Paper 1, you are often asked to simplify expressions like \(2\log_3 x + \log_3 y – \frac{1}{2}\log_3 z\). Combining terms gives \(\log_3 \left(\frac{x^2 \sqrt{y}}{\sqrt{z}}\right)\). Pay attention to the order of operations and use brackets.
在IB AA Paper 1中，常要求化简形如 \(2\log_3 x + \log_3 y – \frac{1}{2}\log_3 z\) 的表达式。合并后得到 \(\log_3 \left(\frac{x^2 \sqrt{y}}{\sqrt{z}}\right)\)。注意运算顺序和括号的书写。
In IB Applications (AI), logs appear in modelling with sinusoidal or logistic functions, and in calculating pH, sound intensity, or Richter scale values. You should be able to switch between exponential and logarithmic forms fluently.
在IB应用与解释（AI）课程中，对数出现在正弦或逻辑函数建模、pH计算、声强或里氏震级中。你需要熟练地在指数形式与对数形式之间转换。
12. Practice and Summary | 练习与总结
Here is a quick summary table of the logarithm rules you must memorise for the IB exam.
下面是对数法则的快速总结表，供IB考试记忆。
| Rule | 法则 | Formula | 公式 |
| Product | 乘积 | \(\log_a (MN) = \log_a M + \log_a N\) |
| Quotient | 商 | \(\log_a \frac{M}{N} = \log_a M – \log_a N\) |
| Power | 幂 | \(\log_a (M^p) = p \log_a M\) |
| Change of base | 换底 | \(\log_a M = \frac{\log_b M}{\log_b a}\) |
| Identity | 恒等式 | \(\log_a a = 1,\ \log_a 1 = 0,\ a^{\log_a x} = x\) |
To reinforce these rules, attempt past paper questions that require simplifying log expressions, solving equations, and using the change of base formula. Pay close attention to the base of the logarithm and the domain of the variable.
为巩固这些法则，请练习历年真题中涉及化简对数表达式、解方程和使用换底公式的题目。密切注意对数的底数和变量的定义域。
Mastering logarithm rules is not just about memorising formulas; it is about understanding the inverse relationship between exponents and logarithms. With consistent practice, you will be able to handle any logarithmic problem confidently in your IB exams.
掌握对数法则不仅仅是记住公式，更是理解指数与对数之间的互逆关系。通过持续练习，你将能够在IB考试中自信地处理任何对数问题。
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