Step by step
The question
Question: work out log₃(81/3).
We use a log rule.
log3(813)
1 Use the quotient rule
The quotient rule: divide inside the log becomes minus outside.
log(ab) = log a − log b
Why?
A log is a power.
81 is 3⁴, and 3 is 3¹.
Dividing takes away the powers.
81 = 3⁴, 3 = 3¹
3431 = 34 − 1
So log₃(81/3) is log₃ 81 minus log₃ 3.
= log3 81 − log3 3
2 Work out each log
log₃ 81 is 4, because 3⁴ is 81.
log₃ 3 is 1, because 3¹ is 3.
3⁴ = 81
3¹ = 3
So we get 4 minus 1.
= 4 − 1
4 minus 1 is 3.
So log₃(81/3) = 3.
log3(813) = 3
3 Check
Let's check: work out the inside first.
81 divided by 3 is 27.
Check: 813 = 27
log₃ 27 is 3, because 3³ is 27.
Same answer!
log3 27 = 3, because 33 = 27
Let's look back.
First: use the quotient rule.
Then: work out each log.
Last: check.
Answer
log3(813) = 3