Just as for the exponents, logarithms have some laws which make working with them easier. These laws are based on the exponential laws and are summarised first and then explained in detail.
Logarithm law 1:
For example,
and
Logarithm law 1:
:
Simplify the following:
Logarithm law 2:
For example,
and
Logarithm law 2:
:
Simplify the following:
Useful to know and remember
When the base is 10, we do not need to state it. From the work done up to now, it is also useful to summarise the following facts:
Logarithm law 3:
The derivation of this law is a bit trickier than the first two. Firstly, we need to relate
and
to the base
. So, assume that
and
. Then from Equation
[link] , we have that:
This means that we can write:
For example, show that
. Start with calculating the left hand side:
The right hand side:
Both sides are equal. Therefore,
.
Logarithm law 3:
:
Write as seperate logs:
Logarithm law 4:
The derivation of this law is identical to the derivation of Logarithm Law 3 and is left as an exercise.
For example, show that
. Start with calculating the left hand side:
The right hand side:
Both sides are equal. Therefore,
.
Logarithm law 4:
:
Write as seperate logs:
Logarithm law 5:
Once again, we need to relate
to the base
. So, we let
. Then,
For example, we can show that
.
Therefore,
.
Logarithm law 5:
:
Simplify the following:
Logarithm law 6:
The derivation of this law is identical to the derivation of Logarithm Law 5 and is left as an exercise.
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