# Logarithms

First some notation....

```
n^x = n to the power of x
```

and

```
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)