Log 计算技巧 Logarithm Calculation Skills

作者：

radmin

在

Chase

Basic Notation & Domain

For a logarithm \(\boldsymbol{\log_b a}\):

b: base; a: argument
Domain restrictions (must be satisfied):\(\boldsymbol{b>0,\ b\neq1,\ a>0}\)

1. Basic Definitions & Core Identities

(1) Conversion between Exponents and Logarithms

\(a^N = b \iff \log_a b = N\)

Usage: Convert between exponential form and logarithmic form.

(2) Special Logarithm Values

\(\log_a 1 = 0,\quad \log_a a = 1\)

The logarithm of 1 is always 0 for any valid base.
The logarithm equals 1 when the base is equal to the argument.

(3) Logarithm Identity

\(a^{\log_a N} = N\)

Usage: Cancel out exponent and logarithm for simplification.

. Change of Base Formula (Most Important Rule)

(1) Standard Formula

\(\log_b a = \frac{\log_c a}{\log_c b} \quad (c>0,\ c\neq1)\)

Usage: Unify different bases to simplify expressions or solve equations.

(2) Three Common Transformations

Reciprocal Identity\(\log_b a = \frac{1}{\log_a b}\)Rule: Swap base and argument, and the logarithm becomes its reciprocal.
Power Transformation\(\log_{b^m} a^n = \frac{n}{m}\log_b a\)Usage: Simplify logarithms with powers or radicals in base/argument.
Chain Cancellation for Product\(\log_{a_1}a_2 \cdot \log_{a_2}a_3 \cdots \log_{a_{k-1}}a_k = \log_{a_1}a_k\)Usage: Middle terms cancel out for continuous product of logarithms.

Arithmetic Rules for Logarithms (Same Base Only)

These rules apply only when logarithms share the same base:

Product rule: \(\log_a(MN) = \log_a M + \log_a N\)
Quotient rule: \(\log_a \dfrac{M}{N} = \log_a M – \log_a N\)
Power rule: \(\log_a M^n = n\log_a M\)
Radical rule: \(\log_a \sqrt[n]{M} = \dfrac{1}{n}\log_a M\)

❌ Common Mistake:

\(\boldsymbol{\log_a(M\pm N) \neq \log_a M \pm \log_a N}\)

Logarithms cannot split addition or subtraction.

4. Quick Calculation Corollaries

\(\log_{\frac{1}{a}} b = -\log_a b\)If the base is the reciprocal, the logarithm changes sign.
\(\log_a b \cdot \log_b a = 1\)The product of two reciprocal logarithms equals 1.
\(x^{\log_y z} = z^{\log_y x}\)Used to simplify complex exponential expressions.

5. Problem-Solving Strategies

Type 1: Logarithmic Equations (Product Form, Same Argument)

Form: \(\log_{f(x)}C \cdot \log_{g(x)}C = \log_{h(x)}C\)

Standard 4-step method:

Use the reciprocal identity to unify all bases to the constant C;
Expand expressions using logarithm addition rules;
Use substitution to convert the equation into a quadratic equation;
For the product/sum of all solutions: apply Vieta’s Formulas directly.
Check roots to satisfy the domain restrictions.

Type 2: Simplify Logarithmic Products

Use change of base formula + chain cancellation to eliminate middle terms.

Type 3: Compare Logarithm Sizes

Use the monotonicity of logarithmic functions and reference values (\(0, 1\)) for comparison.

6. Common Pitfalls

Ignore domain restrictions: Always check roots after solving equations;
Wrongly split \(\log(M\pm N)\);
Miss negative signs or coefficients when simplifying powers/radicals;
Confuse \(\log_b a\) and \(\log_a b\).

前置说明

记对数 \(\boldsymbol{\log_b a}\)：b 为底数，a 为真数；

定义域强制要求：\(\boldsymbol{b>0,\ b\neq1,\ a>0}\)，所有公式均在此前提下成立。

1. 基本定义与核心恒等式（基础必背）

（1）指数与对数互化

\(a^N = b \iff \log_a b = N\)

用途：指数、对数形式互相转换，是所有运算的起点。

（2）特殊对数值（口算秒杀）

\(\log_a 1 = 0,\quad \log_a a = 1\)

任意合法底数，1 的对数为 0，底数与真数相等时对数为 1。

（3）对数恒等式

\(a^{\log_a N} = N\)

用途：指数与对数抵消，常用于化简、求方程解的乘积 / 幂运算。

2. 换底公式（对数核心，竞赛高频）

（1）标准换底公式

\(\log_b a = \frac{\log_c a}{\log_c b} \quad (c>0,c\neq1)\)

作用：统一不同对数的底数，是化简、解方程的核心工具。

（2）三大常用变形（直接套用）

倒数公式（原题核心用法）\(\log_b a = \frac{1}{\log_a b}\)口诀：底数、真数互换，对数互为倒数。
幂次变形（化简大底数 / 高次真数）\(\log_{b^m} a^n = \frac{n}{m}\log_b a\)可快速降次，处理带平方、根式的对数。
链式连乘约分（连乘题型秒杀）\(\log_{a_1}a_2 \cdot \log_{a_2}a_3 \cdots \log_{a_{k-1}}a_k = \log_{a_1}a_k\)连续对数相乘，中间项全部抵消。

3. 同底数对数四则运算法则

仅适用于底数相同的对数，用于展开或合并式子：

积的对数：\(\log_a(MN) = \log_a M + \log_a N\)
商的对数：\(\log_a \dfrac{M}{N} = \log_a M – \log_a N\)
幂的对数：\(\log_a M^n = n\log_a M\)
根式特例：\(\log_a \sqrt[n]{M} = \dfrac{1}{n}\log_a M\)

重要禁忌：\(\boldsymbol{\log_a(M\pm N) \neq \log_a M \pm \log_a N}\)

对数不能拆分加减，仅能拆分乘、除、乘方。

4. 速算二级结论（选择 / 填空专用）

\(\log_{\frac{1}{a}} b = -\log_a b\)（底数互为倒数，对数取相反数）
\(\log_a b \cdot \log_b a = 1\)（互倒对数乘积为 1）
\(x^{\log_y z} = z^{\log_y x}\)（指数对数互换，复杂幂化简）

5. 经典题型解题套路

题型 1：多底数、同真数的对数乘积方程（你所做真题类型）

形式：\(\log_{f(x)}C \cdot \log_{g(x)}C = \log_{h(x)}C\)

通用四步解法：

用倒数公式统一底数为常数 C；
利用对数加法法则展开式子；
换元，转化为一元二次方程；
求所有解的和 / 积：直接用韦达定理，无需单独解方程；最后验根。

题型 2：对数化简 / 连乘

优先使用换底公式 + 链式约分，消去中间项快速得出结果。

题型 3：比较对数大小

结合对数单调性，借助中间值 \(0、1\) 分组判断。

6. 高频易错点

忽略定义域：解方程后必须检验，排除底数≤0、底数 = 1、真数≤0 的增根；
错误拆分 \(\log(M\pm N)\)；
幂次、负号遗漏，根式化简系数写错；
混淆 \(\log_b a\) 与 \(\log_a b\)（二者互为倒数）。