Step by step
The question
Question: write log₂ x + 3 log₂ y − log₂ z as one log.
log2 x + 3 log2 y − log2 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 log₂ y is log₂ of y to the power 3.
Careful: not 3log2 y = log2(3y) ✗
Each number in front goes up as a power: log₂ y³.
= log2 x + log2 y3 − log2 z
2 Plus is times, minus is divide
Plus between logs becomes times inside: x and y³ multiplied.
log a + log b = log(a × b)
Minus becomes divide: z goes to the bottom.
log a − log b = log(ab)
So it is one log: log₂(xy³ over z).
= log2(xy3z)
3 Check
Let's check with numbers: x = 3, y = 2, z = 5.
With a calculator, the question is about 2.263.
x = 3, y = 2, z = 5: question ≈ 2.263
Our answer: the inside is 4.8, and its log is about 2.263.
The same.
It works!
answer: log2 4.8 ≈ 2.263
Let's look back.
First: numbers in front go up.
Then: plus is times, minus is divide.
Last: check.
Answer
= log2(xy3z)