Answer:
[tex]3\,log (x-2)+2\,log(x+2)[/tex] can be written as the single logarithm:
[tex]log((x-2)^3\,(x+2)^2)[/tex]
Step-by-step explanation:
Start by using in both terms of the expression the property of logarithm of a power that states:
[tex]log(a^b)=b\,*\,log(a)[/tex]
so we recognize that the factor that multiplies each logarithmic expression, can be understood as the power of the argument of the function logarithm:
[tex]3\,log(x-2)=log((x-2)^3)\\and\\2\,log(x+2)=log((x+2)^2)[/tex]
Then we use the property of logarithm of a product which states that:
[tex]log(a*b)=log(a)+log(b)[/tex]
So now we recognize the expression:
[tex]log((x-2)^3)\,+\,log((x+2)^2)[/tex]
as the logarithm of the product:
[tex]log((x-2)^3\,(x+2)^2)[/tex]
which is finally a single logarithm as requested.
