Start by using two auxiliary logarithmic expressions. Then rewrite them in exponential form and apply the Properties of Exponents. Finally, go back to logarithmic expressions after simplifying.
See solution.
Practice makes perfect
To derive the Properties of Logarithms, we can start by defining two auxiliary general logarithmic expressions.
a = log_b m c = log_b n
Now, from the definition of logarithm we know that there is an equivalent exponential representation for a logarithmic expression.
y= b^x ⇔ x = log_b y
We can use this to rewrite our auxiliary logarithmic expressions in exponential form.
a = log_b m c = log_b n
⇕
b^a= m b^c= n
This way, we can alter them using the Properties of Exponents. 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.
