Step by step
The question
Question: expand ln(x² over y³).
Expand means: write it as single logs, added and taken away.
ln(x2y3)
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.
= ln x2 − ln y3
2 Bring the powers down
Each power comes down in front, as times: 2 ln x and 3 ln y.
So it is 2 ln x − 3 ln y.
= 2 ln x − 3 ln y
3 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 0.118.
x = 3, y = 2: question ≈ 0.118
Our answer gives about 0.118 too.
It works!
answer ≈ 0.118
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Last: check.
Answer
= 2 ln x − 3 ln y