Logarithm Laws Explained with Worked Examples | 对数运算法则详解与题型归纳

📚 Logarithm Laws Explained with Worked Examples | 对数运算法则详解与题型归纳

Logarithms are the inverse operation of exponentiation. Mastering their laws is essential for solving exponential equations and simplifying expressions in algebra, calculus, and beyond.

对数是指数运算的逆运算。掌握对数运算法则，是解指数方程、化简代数式的重要基础，也是进一步学习微积分等内容的必备工具。

1. Definition and Basic Properties | 定义与基本性质

Let a>0, a≠1 and x>0. The logarithm of x to base a is the exponent y such that a^y = x.

设 a>0，a≠1，x>0。以 a 为底 x 的对数，就是使得 a^y = x 成立的指数 y。

log_a x = y ⇔ a^y = x

• log_a 1 = 0 because a⁰ = 1.

  log_a 1 = 0，因为 a⁰ = 1。

• log_a a = 1 because a¹ = a.

  log_a a = 1，因为 a¹ = a。

• log_a a^y = y, for every real number y.

  log_a a^y = y，对任意实数 y 成立。

• a^{log_a x} = x, where x>0.

  a^{log_a x} = x，其中 x>0。

These four properties follow directly from the definition and are used repeatedly in all logarithmic calculations.

这四个性质直接由定义推出，在后续的所有对数计算中都会反复用到。

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2. Product Rule | 积的对数法则

For M>0 and N>0, the logarithm of a product equals the sum of the logarithms of the individual factors.

当 M>0 且 N>0 时，两个数乘积的对数，等于这两个数的对数之和。