I started out doing this the complicated way, but then I spotted the easy way,
and I realized how awkward and unhelpful my first method was.
log₃ ( 27/81 ) .
Before we even begin to worry about the log , let's reduce that ugly fraction
to lowest terms (simplest form).
Divide top and bottom by 27 :
log₃ (1/3) . Now it's a lot less scary.
Definition of log₃ of (some number):
The power that 3 must be raised to in order to get the number.
What power do you have to raise 3 to in order to get 1/3 ?
Hint: What does ' 3⁻¹ ' mean ?
' 3⁻¹ ' means 1/3 .
So -1 is the power to which you have to raise 3 in order to get 1/3.
So the log₃ of (1/3) is -1 .
I suspect this is going in and out of focus as you read it.
Please go back and read it another 2 or 3 times, until it
snaps into focus.