# What is the value of log2

## Logarithm and its rules of calculation

The logarithm is the inverse of exponentiation. This is an important topic, here you will find an overview of everything that is important, first of all how the logarithm is defined:

log_{b}a = x → b^{x}= a

Spoken this means: "Logarithm from a to base b". Here is ...:

- b the base
- a is the value that comes out if you take b to the power of x
- x is the exponent

### How does the logarithm work?

You need the logarithm to solve equations in which the exponent is unknown, otherwise you would not be able to solve these equations. For example, you want to calculate this x:

2^{x}=1024

Finding this out is not that easy, but you can solve it with the logarithm, which is then:

2^{x}= 1024 -> log_{2}1024 = x

x = 10

**Examples:**

- log
_{2}8=3 → 2^{3}=8 - log
_{3}9=2 → 3^{2}=9 - log
_{3}3=1 → 3^{1}=3

**Exercises with examples:**

Here are tasks that you can calculate or just look at. Click on show to see the solution:

2^{x}+2=10 | Fade in | |

Solution: 2 2 -> log -> x = 3 |

3^{x}+4=13 | Fade in | |

Solution: 3 3 -> log -> x = 2 |

log_{5}x = 2 | Fade in | |

Solution: log → 5 → x = 25 |

So every logarithm is called, which has the 10 as a base. You need this not only for exponents with base 10, but also to calculate other logarithms in the calculator, since most calculators do not have a key for all logarithms. Usually the decadic logarithm is with **lg** abbreviated.

log_{10}(a) = lg (a)

The so-called natural logarithm is a logarithm with a base e (Euler's number). This is a special infinitely non-periodic number (like π too). This logarithm also has a special abbreviation:

log_{e}(x) = ln (x)

### Enter the logarithm in the calculator

To type in a logarithm in the calculator that is neither the decadic nor the natural logarithm, e.g. with the base 2, you need the decadic or natural logarithm. You then divide the natural / decadic logarithm of the number by the natural / decadic logarithm of the base. It doesn't matter whether you use the natural or the decadic logarithm, it just has to be divided into two parts:

Examples:

### Logarithmic Laws - Calculation Rules

*"Product becomes a sum"*

log_{b}(a * c) = log_{b}a + log_{b}c

Example:

log_{3}(x x 9) = log_{3}x + log_{3}9

This rule says that if there is a product in brackets next to the logarithm, you can calculate the logarithm for both factors individually and then add them up.

log_{2}(x 4) | Fade in | |

Solution: → log → log |

log_{3}(9x) | Fade in | |

Solution: → log → log |

*"Division becomes subtraction"*

Example:

log_{3}(x / 9) = log_{3}x-log_{3}9

This rule says that if there is a division or a fraction in brackets, you can do it as with the product, only with a minus.

log_{3}(x / 27) | Fade in | |

Solution: → log → log |

log_{2}(8 / x) | Fade in | |

Solution: → log → 3-log |

*"Exponents can be preferred"*

log_{b}a^{n}= n log_{b}a

Example:

log_{3}9^{2}= 2 x log_{3}9

This rule says that if base (a) has an exponent, you can pull it before the logarithm.

*Division with the same base*

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