Express using a single log 3logab - 2logb - 3loga?

1 Answer
Apr 21, 2018

logblogb

Explanation:

Recall the exponent property for logarithms, which states that bloga=loga^bbloga=logab.

So, 2logb=logb^2, 3loga=loga^3.2logb=logb2,3loga=loga3.

We now have

3logab-logb^2-loga^3=3logab-(logb^2+loga^3)3logablogb2loga3=3logab(logb2+loga3)

The product property for logarithms tells us that loga+logb=logab,loga+logb=logab, so

logb^2+loga^3=log(b^2a^3)logb2+loga3=log(b2a3)

We then get

3logab-logb^2a^33logablogb2a3

Applying the exponent property again,

3logab=log(ab)^3=loga^3b^33logab=log(ab)3=loga3b3

So we are left with

loga^3b^3-logb^2a^3loga3b3logb2a3

The quotient property for logarithms tells us that loga-logb=log(a/b)logalogb=log(ab) so

loga^3b^3-logb^2a^3=log((cancel(a^3)b^cancel(3))/(cancel(a^3)cancel(b^2)))=logb