Step by step
The question
Question: write 3 log x + 2 log y − 4 log z as one log.
3 log x + 2 log y − 4 log z
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 log x is log of x to the power 3.
Careful: not 3log x = log(3x) ✗
Each number in front goes up as a power: log x³, log y² and log z⁴.
= log x3 + log y2 − log z4
2 Plus is times, minus is divide
Plus between logs becomes times inside: x³ and y² multiplied.
log a + log b = log(a × b)
Minus becomes divide: z⁴ goes to the bottom.
log a − log b = log(ab)
So it is one log: log(x³y² over z⁴).
= log(x3y2z4)
3 Check
Let's check with numbers: x = 3, y = 2, z = 5.
With a calculator, the question is about −0.762.
x = 3, y = 2, z = 5: question ≈ −0.762
Our answer: the inside is 0.1728, and its log is about −0.762.
The same.
It works!
answer: log 0.1728 ≈ −0.762
Let's look back.
First: numbers in front go up.
Then: plus is times, minus is divide.
Last: check.
Answer
= log(x3y2z4)