# A-level Mathematics/CIE/Pure Mathematics 2/Logarithmic and Exponential Functions

## Logarithms and Exponents

editA **logarithm** is the inverse function of an exponent.

e.g. The inverse of the function is .

In general, , given that .

## Laws of Logarithms

editThe laws of logarithms can be derived from the laws of exponentiation:

These laws apply to logarithms of any given base

## Natural Logarithms

editThe **natural logarithm** is a logarithm with base , where is a constant such that the function is its own derivative.

The natural logarithm has a special symbol:

The graph exhibits exponential growth when and exponential decay when . The inverse graph is . Here is an interactive graph which shows the two functions as inverses of one another.

## Solving Logarithmic and Exponential Equations

editAn **exponential equation** is an equation in which one or more of the terms is an exponential function. e.g. . Exponential equations can be solved with logarithms.

e.g. Solve

A **logarithmic equation** is an equation wherein one or more of the terms is a logarithm.

e.g. Solve ^{[note 1]}

## Converting Relationships to a Linear Form

editIn maths and science, it is easier to deal with linear relationships than non-linear relationships. Logarithms can be used to convert some non-linear relationships into linear relationships.

### Exponential Relationships

editAn **exponential relationship** is of the form . If we take the natural logarithm of both sides, we get . We now have a linear relationship between and .

e.g. The following data is related with an exponential relationship. Determine this exponential relationship, then convert it to linear form.

x | y |
---|---|

0 | 5 |

2 | 45 |

4 | 405 |

Now convert it to linear form by taking the natural logarithm of both sides:

### Power Relationships

editA **power relationship** is of the form . If we take the natural logarithm of both sides, we get . This is a linear relationship between and .

e.g. The amount of time that a planet takes to travel around the sun (its orbital period) and its distance from the sun are related by a power law. Use the following data^{[1]} to deduce this power law:

Planet | Distance from Sun /10^{6} km |
Orbital Period /days |
---|---|---|

Earth | 149.6 | 365.2 |

Mars | 227.9 | 687.0 |

Jupiter | 778.6 | 4331 |

- References

- ↑ Retrieved from NASA's Planetary Fact Sheet

- Notes

- ↑ is another way of writing