Step by step
The question
Question: expand log₃(27x² over y⁵).
Expand means: write it as single logs, added and taken away.
The three log rules.
Times inside becomes plus outside.
Divide inside becomes minus outside.
A power inside comes down in front.
Careful!
The rules only work for times, divide and powers.
There is no rule for log of (a plus b).
1 Split the times and divide
The top is 27 and x² multiplied: each gets its own log, with plus.
The bottom is divided: its log is taken away.
2 Bring the powers down
Each power comes down in front, as times: 2 log₃ x and 5 log₃ y.
3 Work out the number logs
log₃ 27 is 3, because 3 to the power 3 is 27.
So it is 3 + 2 log₃ x − 5 log₃ y.
4 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 1.845.
Our answer gives about 1.845 too.
It works!
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Then: work out the number logs.
Last: check.