# What is the value of log 1

## Reciprocal theorems of the logarithm

In addition to the **Laws of logarithms** the so-called also help you to calculate with logarithms **Reciprocal rates**. First of all, you should remember what the different parts of the logarithm are called. In the case of the reciprocal of the logarithms, the various variables swap their place and you can quickly lose track of things.

The variable $ \ textcolor {blue} {a} $ becomes ** Base** called the variable $ \ textcolor {black} {b} $

**or something out of date too**

*number**Logarithm*. The question behind the logarithm is: With which number do I have to increase the base $ \ textcolor {blue} {a} $ to get the number $ \ textcolor {black} {b} $?

**What is the reciprocal?**

$ x ~ \ rightarrow \ frac {1} {x} $

### Reciprocal of a logarithm

Logarithms can be calculated by swapping base and number and taking the reciprocal.

$ \ log _ {\ textcolor {blue} {a}} (\ textcolor {black} {b}) ~ = ~ \ frac {1} {\ log _ {\ textcolor {black} {b}} (\ textcolor {blue} {a})} $

This reciprocal rate is a special case of the basic exchange rate. We choose the number as the new basis.

**Basic exchange rate**

In the event that a logarithm for the base $ \ textcolor {blue} {a} $ is unknown, it can be converted into a quotient of two logarithms for any base ($ \ textcolor {green} {c} $).

$ \ log _ {\ textcolor {blue} {a}} (\ textcolor {black} {b}) ~ = ~ \ frac {\ log _ {\ textcolor {green} {c}} (\ textcolor {black} {b} )} {\ log _ {\ textcolor {black} {\ textcolor {green} {c}}} (\ textcolor {blue} {a})} $

So we can use the basic exchange theorem to prove the first reciprocal theorem:

$ \ log_ {a} (b) = \ frac {\ log_ {b} (b)} {\ log_ {b} (a)} $

Since $ \ log_ {b} (b) ~ = ~ 1 $ we get the proposition:

$ \ log_ {a} (b) = \ frac {1} {\ log_ {b} (a)} $

$ \ log_ {8} (2) = \ frac {1} {\ log_ {2} (8)} = \ frac {1} {3} $

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