📚 A - Logarithms in base 10 | A - 以10为底的对数
A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must a base be raised to obtain a certain number?" In IB Mathematics, understanding logarithms — especially those with base 10, known as common logarithms — is crucial for tackling exponential equations, interpreting real‑world scales, and mastering function theory. In this article, we explore the definition, properties, graphs, and applications of base‑10 logarithms to help you excel in your exams.
对数是指数运算的逆运算。它回答这样一个问题：“底数需要升到多少次幂才能得到给定的数？”在 IB 数学中，理解对数 —— 尤其是以 10 为底的常用对数 —— 对于处理指数方程、解读现实世界中的标度以及掌握函数理论至关重要。本文将介绍以 10 为底的对数的定义、性质、图像及其应用，帮助你在考试中取得优异成绩。
1. What is a Logarithm? | 什么是对数？
tive base b (b ≠ 1), the logarithm of a number x with base b is the exponent y such that bʸ s log_b x = y. For instance, 10² = 100, so log₁₀ 100 = 2. The expression log₁₀ 100 is read "log base 10 of 100". Essentially, logarithms 'undo' exponentiation.
对于任意正底数 b（b ≠ 1），以 b 为底 x 的对数是指数 y，满足 bʸ = x。这记作 log_b x = y。例如，10² = 100，因此 log₁₀ 100 = 2。表达式 log₁₀ 100 读作“以 10 为底 100 的对数”。本质上，对数“撤销”了指数运算。
If you have an exponential equation like 10ʸ exponential equations. In IB, you must be comfortable switching between exponential form bʸ = x and logarithmic form log_b x = y.
如果你遇到指数方程例如 10ʸ = 1000，两边取 log₁₀ 得到 y = log₁₀ 1000 = 3。这种转换是解指数方程的基础。在 IB 中，你需要熟练地在指数形式 bʸ = x 与对数形式 log_b x = y 之间切换。
2. Common Logarithms – Base 10 | 常用对数 —— 以 10 为底
Logarithms with base 10 are called common logarithms. On most scientific calculators, the "log" button computes log₁₀. If you see log x without a base written, it is generally understood to mean log₁₀ x. This convention is widely used in science and engineering because our number system is base 10. Common logarithms simplify calculations with very large or very small numbers.
以 10 为底的对数被称为常用对数。在大多数科学计算器上，“log” 按钮计算的就是 log₁₀。如果你看到不带底数的 log x，通常意味着 log₁₀ x。由于我们的数制是十进制，这一约定在科学和工程领域被广泛使用。常用对数能够简化非常大或非常小的数的计算。
ing many zeros, we can work with the exponent 6 or -6. This property is at the heart of logarithmic scales like the Richter scale for earthquakes and the pH scale for acidity, which we will discuss later.
例如，log₁₀(1000000) = 6，而 log₁₀(0.000001) = -6。我们可以用指数 6 或 -6 来代替书写许多个零。这一性质是对数标度（如衡量地震的里氏震级和衡量酸碱性的 pH 值）的核心，稍后我们将予以讨论。
3. Evaluating Common Logarithms | 计算常用对数
e able to estimate such logs between consecutive integers: since 10¹ = 10 and 10² = 100, log₁₀ 50 is between 1 and 2.
当参数正好是 10 的整数次幂时，计算非常简单：log₁₀(10ⁿ) = n。例如，log₁₀(10³) = 3，log₁₀(1/10) = log₁₀(10⁻¹) = -1。如果不是 10 的幂，就会得到无理数结果。比如 log₁₀ 5 约等于 0.6990，意味着 10⁰·⁶⁹⁹⁰ ≈ 5。你应当能够将这类对数估算在两个连续整数之间：由于 10¹ = 10 且 10² = 100，因此 log₁₀ 50 介于 1 和 2 之间。
. Always check if the argument can be written as a rational power of 10 before reaching for a calculator.
在 IB 考试中，你可能需要化简诸如 log₁₀(1000) + log₁₀(0.01) 的表达式。注意到 1000 = 10³，0.01 = 10⁻²，所以和为 3 + (-2) = 1。在动用计算器之前，请先判断参数是否能写成 10 的有理数次幂。
4. Properties of Logarithms (Base 10) | 对数的性质（以 10 为底）
Logarithms satisfy three key properties that are essential for simplifying expressions and solving equations. While these rules hold for any base, we will state them here for base 10. The product rule states that the log of a product is the sum of the logs. The quotient rule turns division into subtraction. The power rule brings an exponent down as a multiplier.
对数满足三条关键性质，对于化简表达式和求解方程至关重要。这些法则对任意底数都成立，这里我们以底数 10 为例。积的法则指出，乘积的对数等于对数的和。商的法则将除法转化为减法。幂的法则将指数“拿下来”成为乘数。
| Property | Formula (Base 10) |
| Product Rule | log₁₀(mn) = log₁₀ m + log₁₀ n |
| Quotient Rule | log₁₀(m/n) = log₁₀ m - log₁₀ n |
| Power Rule | log₁₀(mⁿ) = n log₁₀ m |
0) = 1. The power rule is especially useful when the argument contains an exponent: log₁₀(√10) = log₁₀(10¹/²) = (1/2) log₁₀ 10 = 0.5.
这些性质使我们能够将复杂的对数表达式分解为更简单的部分。例如，log₁₀(2) + log₁₀(5) = log₁₀(10) = 1。当参数含有指数时，幂的法则尤为有用：log₁₀(√10) = log₁₀(10¹/²) = (1/2) log₁₀ 10 = 0.5。
5. Change of Base Formula | 换底公式
Although our focus is base 10, you will often encounter logarithms with other bases, such as log₂ 8 or log₅ 25. The change of base formula connects any logarithm to common logarithms (or natural logs), making calculations possible on a standard calculator.
尽管我们关注的是以 10 为底的对数，但你经常会碰到其他底数的对数，如 log₂ 8 或 log₅ 25。换底公式可将任意对数与常用对数（或自然对数）联系起来，使得标准计算器能够完成计算。
logₐ b = log₁₀ b / log₁₀ a
For example, to evaluate log₂ 5 using base‑ invaluable when solving exponential equations where the base is not 10 and you need a decimal approximation.
例如，计算 log₂ 5 时，可借助以 10 为底的对数：计算 log₁₀ 5 ÷ log₁₀ 2 ≈ 0.6990 ÷ 0.3010 = 2.3219。当解底数不是 10 的指数方程且需要小数近似值时，该公式极为有用。
6. Solving Exponential Equations with Common Logs | 使用常用对数解指数方程
ndard strategy. If the equation contains base 10, the solution is direct. For instance, 10ˣ = 7 implies x = log₁₀ 7. For a non‑10 base, apply the power rule after taking log₁₀ of both sides.
当未知数出现在指数位置时，标准策略是对两边取对数。如果方程中含有 10 的幂，解法直接。例如，10ˣ = 7 意味着 x = log₁₀ 7。对于底数不是 10 的情形，在两边取 log₁₀ 后再运用幂的法则。
Consider 2ˣ = 30. Taking log₁₀ both sides gives log₁₀(2ˣ 907. Always remember that taking logs of both sides is valid because log functions are one‑to‑one.
考虑 2ˣ = 30。两边取 log₁₀ 得 log₁₀(2ˣ) = log₁₀ 30 → x log₁₀ 2 = log₁₀ 30 → x = log₁₀ 30 / log₁₀ 2。用计算器可求得 x ≈ 4.907。始终记住，由于对数函数是一一对应的，对两边取对数是有效的。
7. Solving Logarithmic Equations | 解对数方程
the left side: log₁₀[x(x - 3)] = 1. Then rewrite in exponential form: x(x - 3) = 10¹ = 10.
对数方程常常需要先用积的法则或商的法则将对数合并，再转化为指数形式。一个典型的 IB 题目是：解方程 log₁₀ x + log₁₀(x - 3) = 1。首先，压缩左端：log₁₀[x(x - 3)] = 1。然后改写为指数形式：x(x - 3) = 10¹ = 10。
valid solution is x = 5. Always check the domain of the original logarithmic expressions.
解二次方程 x² - 3x - 10 = 0 → (x - 5)(x + 2) = 0 → x = 5 或 x = -2。然而，log₁₀ x 要求 x > 0，log₁₀(x - 3) 要求 x > 3。因此 x = -2 是增根，必须舍去。唯一有效的解是 x = 5。务必检验原对数表达式的定义域。
8. Graph of y = log₁₀ x | y = log₁₀ x 的图像
The function f(x) = log₁₀ x is the inverse of g(x) = 10ˣ 0.1, -1). The domain is x > 0, and the range is all real numbers. As x → 0⁺, log₁₀ x → -∞.
函数 f(x) = log₁₀ x 是 g(x) = 10ˣ 的反函数。它的图像以 x = 0 为垂直渐近线，且过点 (1, 0)，因为 log₁₀ 1 = 0。它还经过点 (10, 1) 和 (0.1, -1)。定义域为 x > 0，值域为全体实数。当 x → 0⁺ 时，log₁₀ x → -∞。
g but at a decreasing rate. Comparing y = log₁₀ x with y = ln x (base e ≈ 2.718), you will
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