Step by step
The question
Question: write 4 log₃ x + log₃ 2 as one log.
4 log3 x + log3 2
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 4 in front is a power, not times.
4 log₃ x is log₃ of x to the power 4.
Careful: not 4log3 x = log3(4x) ✗
Each number in front goes up as a power: log₃ x⁴.
= log3 x4 + log3 2
2 Plus becomes times
Plus between logs becomes times inside: 2 and x⁴ multiplied, the number first.
log a + log b = log(a × b)
So it is one log: log₃(2x⁴).
= log3(2x4)
3 Check
Let's check with numbers: x = 3.
With a calculator, the question is about 4.631.
x = 3: question ≈ 4.631
Our answer: the inside is 162, and its log is about 4.631.
The same.
It works!
answer: log3 162 ≈ 4.631
Let's look back.
First: numbers in front go up.
Then: plus becomes times.
Last: check.
Answer
= log3(2x4)