Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Types of Logaritms

对数的类型

Although a logarithm may be defined with any base, the logs most often encountered are the logarithm to the base 10 which is called the common logarithm

尽管对数可以以任何底数定义，但最常遇到的对数是底数为10的对数，称为常用对数

and the logarithm to the base e which is called the natural logarithm.

以及以e为底的对数，称为自然对数。

Base changes can be accomplished. For a chosen base:

基底变换可以实现。对于选定的基底：

Logarithms may be manipulated with the combination rules.

对数可以使用组合规则进行操作。
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Logarithm Combinations

对数组合

The logarithm to the base b of the variable x is defined as the power to which you would raise b to get x. It is written

以b为底的变量x的对数定义为将b提升到某次幂以得到x的次数。它写作
.

The general combination properties of logarithms are:

对数的通用组合性质为：
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Logarithm Base Change

对数换底公式

If the logarithm to the base a is known, then the logarithm to the base b can be obtained by the base change relationship:

如果已知以a为底的对数，那么可以通过底数变换关系得到以b为底的对数：

This can be proved from the definition and combination rules for logarithms. If

这可以从对数的定义和组合规则中证明。如果

then

然后

and rearranging gives

并重新排列得到

The most common base changes are from the natural log to base 10 log or vice versa.

最常见的底数转换是从自然对数转换为底数10的对数，或反之。
log10x = ln x/ln 10 = ln x/2.30258 = 0.4343 ln x
log₁₀x = ln x / ln 10 = ln x / 2.30258 = 0.4343 ln x
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What is a Logarithm?

什么是对数？

The logarithm to the base b of the variable x is defined as the power to which you would raise b to get x. If the logarithm to the base b of x is equal to y, then b raised to the y power will give you the value x. It it written

以b为底的变量x的对数定义为，你需要将b提升到多少次方才能得到x。如果x的以b为底的对数等于y，那么将b提升到y次方就会得到x。它写成
.

The general combination properties of logarithms are:

对数的通用组合性质为：
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