To derive the Properties of Logarithms, we can start by defining two auxiliary general logarithmic expressions.
a=logbmc=logbn
Now, from the definition of we know that there is an equivalent exponential representation for a logarithmic expression.
y=bx⇔x=logby
We can use this to rewrite our auxiliary logarithmic expressions in exponential form.
a=logbmc=logbn⇕ba=mbc=n
This way, we can alter them using the . Once we have simplified by using the Properties of Exponents, we can rewrite our result using the corresponding logarithmic expressions. We will now work out some examples.
Deriving the Product Property of Logarithms
Using the expressions introduced before, lets find the product of
m and
n.
mn=(ba)(bc)
▼
Simplify and rewrite using logarithms
logb(mn)=logbm+logbn
This last result is known as the .
Deriving the Quotient Property of Logarithms
Using the expressions introduced above, we will now find the quotient of
m and
n.
nm=bcba
▼
Simplify and rewrite using logarithms
logb(nm)=logbm−logbn
This last result is known as the .
Deriving the Power Property of Logarithms
Let's consider the logarithm base
b of
mn. Since we know that
m=ba, we can use it in our expression.
logb(mn)=logb((ba)n)
▼
Simplify and rewrite using logarithms
logb(mn)=logb(ban)
logb(mn)=an
logb(mn)=(logbm)n
logb(mn)=nlogbm
This last result is known as the .