## 1. Introduction

Powers and logarithms provide a powerful way of representing large and small quantities, and performing complex calculations. Understanding powers will allow you to make better 'back-of-the-envelope' calculations or to quality-check results from your calculator.

*You can also download the complete web document as a printable PDF file here.*

First, look at the number sequence:

10

100

1 000

10 000

100 000

1 000 000

Each of these numbers is the previous number multiplied by 10. This list can be re-written as:

10 | = | 10 | 1 |

100 | = | 10 × 10 | 2 |

1 000 | = | 10 × 10 × 10 | 3 |

10 000 | = | 10 × 10 × 10 × 10 | 4 |

100 000 | = | 10 × 10 × 10 × 10 × 10 | 5 |

1 000 000 | = | 10 × 10 × 10 × 10 × 10 × 10 | 6 |

The numbers in the right-hand column are the number of multiples of 10 for each number. This is the **power** of 10 and can be added as a superscript to 10 to represent each number:

10 | = | 10 | 10^{1} |

100 | = | 10 × 10 | 10^{2} |

1 000 | = | 10 × 10 × 10 | 10^{3} |

10 000 | = | 10 × 10 × 10 × 10 | 10^{4} |

100 000 | = | 10 × 10 × 10 × 10 × 10 | 10^{5} |

1 000 000 | = | 10 × 10 × 10 × 10 × 10 × 10 | 10^{6} |

*HINT: Count the zeros after the 'one' in the left-hand column - this equals the power*

This pattern continues below 10:

1 | = | 10 | 10^{0} |

0.1 | = | 10 ÷ 10 | 10^{-1} |

0.01 | = | 10 ÷ 10 ÷ 10 | 10^{-2} |

0.001 | = | 10 ÷ 10 ÷ 10 ÷ 10 | 10^{-3} |

0.0001 | = | 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 | 10^{-4} |

0.00001 | = | 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 ÷ 10 | 10^{-5} |

*
HINT: Count the number of zeros after the decimal place and before the first 'one' in the left-hand column, then add one for the decimal point - this equals the 'minus' power.
Note that minus power indicates a value that is the reciprocal of the positive power, thus:
*

Note that minus power indicates a value that is the reciprocal of the positive power, thus:

10^{-3} = 0.001 = 1 ÷ 1000 = 1 ÷ 10^{3}