
Introduction to Logarithms (Logs)
Logarithms or logs are another way of expressing indices and is a bit like the inverse of dealing with powers. Logs were used to perform difficult calculations before there were calculators.
To understand how logs work we could just look at a simple to understand example:
We know that , so if we wanted to know the value of the indices that would make
into
(how many times
multiplied by itself gives us
i.e.
) then we could use the above to work it out. We would say ‘Log to the base
of
is
, or other words,
squared is
‘.
If we look at the original expression and transform that into Log language (how you would input it into your calculator) we’d say ‘LOG TO THE BASE of the number that has a power, OF the answer that we’d get from the indices IS the power’.
Logs can’t be done with negative numbers and always have to be done with positive numbers. If we’ve got a value for a negative log then that is for a value for an indices that is less than i.e.
because when we deal with indices, if we’ve got a negative power that means we’ve got a fraction.
Log in base 10 (Decimal)
Logs work for all powers of and not just integers:
Therefore:



If we look at a comparison (shown in the picture above) of &
we’ll see that the curve of the Red line (
) is very steep at the begining but falls off after
(in this instance) below the Green line (
).
ln – base e
Natural (Napierian) logs work in exactly the same way as base 10 logs but because we are dealing with natural number e we can use ln:
Just like base 10 logs
Rules of logs
There are three rules or laws of logs that can be used in all bases:
Rule 1 Proof
Let ,
,
// Substitute in
&
with
and
Rule 2 Proof
Let ,
,
// Substitute in
&
with
and
Rule 3 Proof
//
times
//
times
Cahnging bases
Logs can be done in any base but calculators only have the option for base 10 or base e so if we need to calculate a different base then we would need to convert the base we want into one fo these two first.
As an example let’s evaluate .
Firstly we would rearrange to get and then take logs (base 10 or e)
. Then simply rearrange for
to get
.