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 BigO 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 BigO term.
This is the reason that the bases of logarithms don’t matter in the BigO notation.