Step by step
The question
Question: write 2 log x − 3 log y as one log.
2 log x − 3 log 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 2 in front is a power, not times.
2 log x is log of x to the power 2.
Careful: not 2log x = log(2x) ✗
Each number in front goes up as a power: log x² and log y³.
= log x2 − log y3
2 Plus is times, minus is divide
Minus becomes divide: y³ goes to the bottom.
log a − log b = log(ab)
So it is one log: log(x² over y³).
= log(x2y3)
3 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 0.051.
x = 3, y = 2: question ≈ 0.051
Our answer: the inside is 1.125, and its log is about 0.051.
The same.
It works!
answer: log 1.125 ≈ 0.051
Let's look back.
First: numbers in front go up.
Then: plus is times, minus is divide.
Last: check.
Answer
= log(x2y3)