Step by step
The question
Question: expand log₃(x/y).
Expand means: write it as single logs, added and taken away.
log3(xy)
The three log rules.
Times inside becomes plus outside.
Divide inside becomes minus outside.
A power inside comes down in front.
log(a × b) = log a + log b
log(ab) = log a − log b
log(an) = n × log a
Careful!
The rules only work for times, divide and powers.
There is no rule for log of (a plus b).
Careful: log(a + b) ≠ log a + log b
1 Split the times and divide
The bottom is divided: its log is taken away.
So it is log₃ x − log₃ y.
= log3 x − log3 y
2 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 0.369.
x = 3, y = 2: question ≈ 0.369
Our answer gives about 0.369 too.
It works!
answer ≈ 0.369
Let's look back.
First: split the times and divide.
Last: check.
Answer
= log3 x − log3 y