Step by step
The question
Question: expand log₂(8√x).
Expand means: write it as single logs, added and taken away.
The three log rules.
Times inside becomes plus outside.
Divide inside becomes minus outside.
A power inside comes down in front.
Careful!
The rules only work for times, divide and powers.
There is no rule for log of (a plus b).
1 Split the times and divide
The top is 8 and √x multiplied: each gets its own log, with plus.
2 Bring the powers down
A square root is the power one half: √x is x to the power 1/2.
Each power comes down in front, as times: 1/2 log₂ x.
3 Work out the number logs
log₂ 8 is 3, because 2 to the power 3 is 8.
So it is 3 + 1/2 log₂ x.
4 Check
Let's check with numbers: x = 3.
With a calculator, the question is about 3.792.
Our answer gives about 3.792 too.
It works!
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Then: work out the number logs.
Last: check.