Notes

Get a Grip on the Logarithm

Daniel Weibel
Created 28 Aug 2017
Last updated 20 Jan 2018

This note explains how the logarithm works, and is supposed to serve as a memory aid for remembering how the logarithm works.

Introduction

The logarithm to the base $b$ of a number $n$ is defined as follows:

If we need to know the logarithm $\Log_b$ of $n$, we can say:

  • The base $b$ to the power of what ($x$) equals $n$?

Or in other words:

  • The logarithm $\Log_b$ of $n$ is the exponent $x$ to which the base $b$ must be raised in order to result in $n$.

Examples

Last example: $e$ is Euler’s number $e = 2.71828$

Logarithm vs. Exponentiation

The logarithm is the inverse operation of exponentiation.

Exponentiation

  • Input: a base and an exponent
  • Output: a number

Examples:

$\mr{base} = 2$ and $\mr{exponent}=5$

  • $2^5 = 32$

$\mr{base} = 10$ and $\mr{exponent}=3$

  • $10^3 = 1000$

Logarithm

  • Input: a base and a number
  • Output: an exponent

Examples:

$\mr{base} = 2$ and $\mr{number}=32$

  • $\Log_2 32 = 5$

$\mr{base} = 10$ and $\mr{number}=1000$

  • $\Log_{10} 1000 = 3$

Convert Between Different Bases

Suppose we know the logarithm $\Log_{b_1}$ of $n$:

Now we want to know the logarithm $\Log_{b_2}$ of $n$. That is, the logarithm of the same number, but to a different base:

We can calculate $x_2$ as follows:

That is:

  • The logarithm $\Log_{b_1} n$, that we already know, divided by the logarithm of the new base $b_2$ to the original base $b_1$

Examples

  • Convert $\Log_2 128 = 7$ from base $2$ to base $10$:

  • Convert $\Log_e 512 = 6.238325$ from base $e$ to base 2:

  • Convert $\Log_e 1000 = 6.907755$ from base $e$ to base 10:

Note Regarding Big-O Notation

Since $\Log_{b_2} n = \frac{\Log_{b_1} n}{\Log_{b_1} b_2}$, the logarithms of different bases differ by the following factor:

This is a constant factor, because it does not contain the input number $n$, which means that it can be drop from the Big-O term.

This is the reason that the bases of logarithms don’t matter in the Big-O notation.