Logarithms

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.

First we add a variable (x) to the expression on the right hand:
$$ \log _{2} 8 \leftrightarrow x
$$
Next we take the base of the logarithm and combine it with x as an exponent. Now our formula looks like this:
$$ \log _{2} 8 \leftrightarrow 2^x
$$
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