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


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


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)

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