eolas/zk/Logarithms.md

---
tags: [algebra]
---

Most simply a logarithm is a way of answering the question:

> How many of one number do we need to get another number. How many of x do we
> need to get y

More formally:

> x raised to what power gives me y

Below is an example of a logarithm:

$$
\log \_{3} 9


$$

We read it:

> log base 3 of 9

And it means:

> 3 raised to what power gives me 9?

In this case the answer is easy: $3^2$ gives me nine, which is to say: three
multiplied by itself.

## Using exponents to calculate logarithms

This approach becomes rapidly difficult when working with larger numbers. It's
not as obvious what $\log \_{5} 625$ would be using this method. For this
reason, we use exponents which are intimately related to logarithms.

A logarithm can be expressed identically using exponents for example:

$$
\log \_{3} 9 = 2 \leftrightarrow 3^2 = 9


$$

By carrying out the conversion in stages, we can work out the answer to the
question a logarithm poses.

Let's work out $\log \_{2} 8$ using this method.

1. First we add a variable (x) to the expression on the right hand:

   $$
   \log \_{2} 8 \leftrightarrow x


   $$

1. Next we take the base of the logarithm and combine it with x as an exponent.
   Now our formula looks like this:

   $$
   \log \_{2} 8 \leftrightarrow 2^x


   $$

1. Next we add an equals and the number that is left from the logarithm (8):

$$
\log \_{2} 8 \leftrightarrow 2^x = 8


$$

Then the problem is reduced to: how many times do you need to multiply two by
itself to get 8? The answer is 3 : 2 x 2 x 2 or 2 p3. Hence we have the balanced
equation:

$$
\log \_{2} 8 \leftrightarrow 2^3 = 8


$$

## Common base values

Often times a base won't be specified in a log expression. For example:

$$
\log1000


$$

This is just a shorthand and it means that the base value is ten, i.e that the
logarithm is written in denary (base 10). So the above actually means:

$$
\log \_{10} 1000 = 3


$$

This is referred to as the **common logarithm**

Another frequent base is Euler's number (approx. 2.71828) known as the **natural
logarithm**

An example

$$
\log \_{e} 7.389 = 2


$$