## Logarithms

``````Log_n (n^x) = x   (where the _n denotes "subscript n" and means the Log was in base n)
``````

For example:

``````Log_10 (10)      = 1   (The subscript 10 means the Log was in base 10)
Log_10 (1000) = 3
Log_2  (32)   = 5
``````

Similarly:

``````Log_e (e) = 1
``````

Log base e is referred to as "the natural log" and written as the function "ln" (pronounced “Lin”)

Where e is Euler's numbers

Hence

``````Log_e (x) = ln(x)   -- much easier to write, no need for subscripts.
``````

By convention, if you write Log(x) without specifying a base, then you assume it to be base 10.

``````Log(1) = 0
Ln(1) = 0

Because n^0 = 1
``````

## Log is undefined for 0 and negative numbers.

As a positive number get smaller and smaller and closer to zero the Log of the number becomes a huge negative number

e.g. 10^-10000000 is 0.0000001

So Log(0.00000001) = -7

What happens when the number reaches 0? We are in spooky 'undefined' territory

``````ln(0) = undefined

ln(-3) undefined
``````

## Basic log rules(these work for any base)

``````Log(m^n) = n Log(m)

Log(a) + Log(b) = Log(a*b)
``````

e.g. Log(10) + Log(1000) = 1+3 = 4 Log(10000) = 4 10000 = 10*1000

``````Log(a) - Log(b) = Log(a/b)
``````

e.g. Log(10000) - Log(100) = Log(10000/100) = Log(100) = 2

The Log base switch rule

``````log_b(c) = 1 / log_c(b)
``````

## Convert to the natural log.

Sometimes you'll have an equation that has a base other than 10 or `e`. To be able to get an answer on your calculator you'll need to convert it to base 10 or base `e`.

In such cases you use: the log base change rule

``````log_b(x) = log_c(x) / log_c(b)
``````

So perhaps you've ended up with an answer of: `log_12 (14)`

To turn this into a number....

``````log_12 (14) = ln(14) / ln(12)
``````

Which you can plug into a calculator (provided it has a `ln` button, i.e. it is a 'scientific' calculator)