# Lovely Logarithms!

## Here's a function:

## y = log_{x}(2x)

## If *x* is equal to a power of ½, then *y* = (*n* - 1)/*n*! (*n* is the exponent that you raise *x* by)

### Note: *n* ≠ 0 because division by zero(0) is undefined.

x | y |

1/2 | 0 |

1/4 | 1/2 |

1/8 | 2/3 |

1/16 | 3/4 |

1/32 | 4/5 |

1/64 | 5/6 |

1/128 | 6/7 |

1/256 | 7/8 |

1/512 | 8/9 |

1/1024 | 9/10 |

½^n | (n - 1)/n |

Since **½** is the reciprocal of **2**, the negative powers of **½** are the positive powers of **2** & *vice versa!*

## If *x* is equal to a power of 2, then *y* = (*n* + 1)/*n*!

### Note: Again, *n* ≠ 0 because division by zero(0) is undefined.

x | y |

2 | 2 |

4 | 3/2 |

8 | 4/3 |

16 | 5/4 |

32 | 6/5 |

64 | 7/6 |

128 | 8/7 |

256 | 9/8 |

512 | 10/9 |

1024 | 11/10 |

2^n | (n + 1)/n |

Also, remember this fact about fractions & mixed numbers:

You can also see this fact on the Main Page of the Math Section.

## Technically, 1 is the zeroth power of 2 & also the zeroth power of ½, but if *x* = 1, then *y* is undefined because log_{1}T is indeterminate. When the number 1 is the base of a logarithm, it's like dividing by zero(0)! Besides, 1^{T} = 1, every time!

Also, notice how I didn't use the exclamation mark as the** factorial function! **

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© Derek Cumberbatch