Step by step
The question
Question: write 3 ln x − ln y as one log.
3 ln x − ln y
The three log rules, used backwards.
A number in front goes up as a power.
Plus becomes times.
Minus becomes divide.
n × log a = log(an)
log a + log b = log(a × b)
log a − log b = log(ab)
1 Numbers in front go up
Careful!
The 3 in front is a power, not times.
3 ln x is ln of x to the power 3.
Careful: not 3ln x = ln(3x) ✗
Each number in front goes up as a power: ln x³.
= ln x3 − ln y
2 Plus is times, minus is divide
Minus becomes divide: y goes to the bottom.
log a − log b = log(ab)
So it is one log: ln(x³ over y).
= ln(x3y)
3 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 2.603.
x = 3, y = 2: question ≈ 2.603
Our answer: the inside is 13.5, and its log is about 2.603.
The same.
It works!
answer: ln 13.5 ≈ 2.603
Let's look back.
First: numbers in front go up.
Then: plus is times, minus is divide.
Last: check.
Answer
= ln(x3y)