Step by step
The question
Question: expand log(100x³).
Expand means: write it as single logs, added and taken away.
log(100x3)
The three log rules.
Times inside becomes plus outside.
Divide inside becomes minus outside.
A power inside comes down in front.
log(a × b) = log a + log b
log(ab) = log a − log b
log(an) = n × log a
Careful!
The rules only work for times, divide and powers.
There is no rule for log of (a plus b).
Careful: log(a + b) ≠ log a + log b
1 Split the times and divide
The top is 100 and x³ multiplied: each gets its own log, with plus.
= log 100 + log x3
2 Bring the powers down
Each power comes down in front, as times: 3 log x.
= log 100 + 3 log x
3 Work out the number logs
log 100 is 2, because 10 to the power 2 is 100.
So it is 2 + 3 log x.
= 2 + 3 log x
102 = 100
4 Check
Let's check with numbers: x = 3.
With a calculator, the question is about 3.431.
x = 3: question ≈ 3.431
Our answer gives about 3.431 too.
It works!
answer ≈ 3.431
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Then: work out the number logs.
Last: check.
Answer
102 = 100