# rigorous definition of the logarithm

In this entry, we shall construct the logarithm as a Dedekind cut and then demonstrate some of its basic properties. All that is required in the way of background material are the properties of integer powers of real numbers.

###### Theorem 1.

Suppose that $a\mathrm{,}b\mathrm{,}c\mathrm{,}d$ are positive integers such that $a\mathrm{/}b\mathrm{=}c\mathrm{/}d$ and that $x\mathrm{>}\mathrm{0}$ and $y\mathrm{>}\mathrm{0}$ are real numbers. Then ${x}^{a}\mathrm{\le}{y}^{b}$ if and only if ${x}^{c}\mathrm{\le}{y}^{d}$.

###### Proof.

Cross multiplying, the condition $a/b=c/d$ is equivalent^{} to $ad=bc$.
By elementary properties of powers, ${x}^{a}\le {y}^{b}$ if and only if
${x}^{ad}\le {y}^{bd}$. Likewise, ${x}^{c}\le {x}^{d}$ if and only if ${x}^{bc}\le {y}^{bd}$ which, since $bc=ad$, is equivalent to ${x}^{ad}\le {y}^{bd}$.
Hence, ${x}^{a}\le {y}^{b}$ if and only if ${x}^{c}\le {x}^{d}$.
∎

###### Theorem 2.

Suppose that $a\mathrm{,}b\mathrm{,}c\mathrm{,}d$ are positive integers such that $a\mathrm{/}b\mathrm{\le}c\mathrm{/}d$ and that $x\mathrm{>}\mathrm{1}$ and $y\mathrm{>}\mathrm{0}$ are real numbers. If ${x}^{c}\mathrm{\le}{y}^{d}$ then ${x}^{a}\mathrm{\le}{y}^{b}$.

###### Proof.

Since we assumed that $b>0$, we have that ${x}^{c}\le {y}^{d}$ is equivalent to ${x}^{bc}\le {y}^{bd}$. Likewise, since $d>0$, we have that ${x}^{a}\le {y}^{b}$ is equivalent to ${x}^{ad}\le {y}^{bd}$. Cross-multiplying, $a/b\le c/d$ is equivalent to $ad\le bc$. Since $x>1$, we have ${x}^{ad}\le {x}^{bc}$. Combining the above statements, we conclude that ${x}^{c}\le {y}^{d}$ implies ${x}^{a}\le {y}^{b}$. ∎

###### Theorem 3.

Suppose that $a\mathrm{,}b\mathrm{,}c\mathrm{,}d$ are positive integers such that $a\mathrm{/}b\mathrm{>}c\mathrm{/}d$ and that $x\mathrm{>}\mathrm{1}$ and $y\mathrm{>}\mathrm{0}$ are real numbers. If ${x}^{a}\mathrm{>}{y}^{b}$ then ${x}^{c}\mathrm{>}{y}^{d}$.

###### Proof.

Since we assumed that $b>0$, we have that ${x}^{c}>{y}^{d}$ is equivalent to ${x}^{bc}>{y}^{bd}$. Likewise, since $d>0$, we have that ${x}^{a}>{y}^{b}$ is equivalent to ${x}^{ad}>{y}^{bd}$. Cross-multiplying, $a/b>c/d$ is equivalent to $ad>bc$. Since $x>1$, we have ${x}^{ad}>{x}^{bc}$. Combining the above statements, we conclude that ${x}^{c}>{y}^{d}$ implies ${x}^{a}>{y}^{b}$. ∎

###### Theorem 4.

Let $x\mathrm{>}\mathrm{1}$ and $y$ be real numbers. Then there exists an integer $n$ such that ${x}^{n}\mathrm{>}y$.

###### Proof.

Write $x=1+h$. Then we have ${(1+h)}^{n}\ge 1+nh$ for all
positive integers $n$. This fact is easily proved by induction^{}.
When $n=1$, it reduces to the triviality $1+h\ge h$. If
${(1+h)}^{n}\ge 1+nh$, then

$${(1+h)}^{n+1}=(1+h){(1+h)}^{n}\ge (1+h)(1+nh)=1+(n+1)h+n{h}^{2}\ge 1+(n+1)h.$$ |

By the Archimedean property, there exists an integer $n$ such that $1+nh>y$, so ${x}^{n}>y$. ∎

###### Theorem 5.

Let $x\mathrm{>}\mathrm{1}$ and $y$ be real numbers. Then the pair of sets $\mathrm{(}L\mathrm{,}U\mathrm{)}$ where

$L$ | $=\{r\in \mathbb{Q}\mid (\exists a,b\in \mathbb{Z})\mathit{\hspace{1em}}b>0\wedge r=a/b\wedge {x}^{a}\le {y}^{b}\}$ | (1) | ||

$U$ | $=\{r\in \mathbb{Q}\mid (\exists a,b\in \mathbb{Z})\mathit{\hspace{1em}}b>0\wedge r=a/b\wedge {x}^{a}>{y}^{b}\}$ | (2) |

forms a Dedekind cut.

###### Proof.

Let $r$ be any rational number^{}. Then we have $r=a/b$ for some integers
$a$ and $b$ such that $b>0$. The possibilities ${x}^{a}\le {y}^{b}$ and
${x}^{a}>{y}^{b}$ are exhaustive so $r$ must belong to at least one of $U$ and
$L$. By theorem 1, it cannot belong to both. By theorem 2, if $r\in L$
and $s\le r$, then $s\in L$ as well. By theorem 3, if $r\in U$
and $s>r$, then $s\in U$ as well. By theorem 4, neither $L$ nor $U$
are empty. Hence, $(L,U)$ is a Dedekind cut and defines a
real number.
∎

###### Definition 1.

Suppose $x\mathrm{>}\mathrm{1}$ and $y\mathrm{>}\mathrm{0}$ are real numbers. Then, we define ${\mathrm{log}}_{x}\mathit{}y$ to be the real number defined by the cut $\mathrm{(}L\mathrm{,}U\mathrm{)}$ of the above theorem.

Title | rigorous definition of the logarithm |
---|---|

Canonical name | RigorousDefinitionOfTheLogarithm |

Date of creation | 2013-03-22 17:00:37 |

Last modified on | 2013-03-22 17:00:37 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 18 |

Author | rspuzio (6075) |

Entry type | Derivation |

Classification | msc 26A06 |

Classification | msc 26A09 |

Classification | msc 26-00 |