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

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Answer 1 · 2018-04-21T05:04:11

#logb#

Recall the exponent property for logarithms, which states that #bloga=loga^b#.

So, #2logb=logb^2, 3loga=loga^3.#

We now have

#3logab-logb^2-loga^3=3logab-(logb^2+loga^3)#

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

#logb^2+loga^3=log(b^2a^3)#

We then get

#3logab-logb^2a^3#

Applying the exponent property again,

#3logab=log(ab)^3=loga^3b^3#

So we are left with

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

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

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