Logarithmic Base Change Formula and Derivation Proof

1. Basic concepts

In mathematics, logarithms are the inverse of exponentiation, just as division is the inverse of multiplication and vice versa. If the power aof xis equal to N(a>0, and a≠1), then the number xis acalled Nthe base logarithm (logarithm), recorded as $x=lo{g}_{a}N.$ _ Among them,acalled the base of the logarithm,Ncalled the real number. $x=lo{g}_{a}N$等了于$a^x=N$。

• In particular, we call the base 10 logarithm the common logarithm (common logarithm), and denote it as lgN.
• The logarithm e(e=2.71828…)based on an irrational number is called the natural logarithm (natural logarithm), and is recorded as lnN.
• Zero has no logarithm, because no number can be raised to a power of 0.
• In the range of real numbers, negative numbers have no logarithms. In the range of imaginary numbers, negative numbers have logarithms.

The image of a logarithm is as follows:

Common properties of logarithms:

2. Bottom change formula