Worked examples
The question
Question: expand log(x³y).
Expand means: write it as single logs, added and taken away.
log(x3y)
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 x³ and y multiplied: each gets its own log, with plus.
= log x3 + log y
2 Bring the powers down
Each power comes down in front, as times: 3 log x.
So it is 3 log x + log y.
= 3 log x + log y
3 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 1.732.
x = 3, y = 2: question ≈ 1.732
Our answer gives about 1.732 too.
It works!
answer ≈ 1.732
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Last: check.
▶ Watch it solved
The question
Question: expand log₃(9x² over y).
Expand means: write it as single logs, added and taken away.
log3(9x2y)
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 9 and x² multiplied: each gets its own log, with plus.
The bottom is divided: its log is taken away.
= log3 9 + log3 x2 − log3 y
2 Bring the powers down
Each power comes down in front, as times: 2 log₃ x.
= log3 9 + 2 log3 x − log3 y
3 Work out the number logs
log₃ 9 is 2, because 3 to the power 2 is 9.
So it is 2 + 2 log₃ x − log₃ y.
= 2 + 2 log3 x − log3 y
32 = 9
4 Check
Let's check with numbers: x = 3, y = 2.
With a calculator, the question is about 3.369.
x = 3, y = 2: question ≈ 3.369
Our answer gives about 3.369 too.
It works!
answer ≈ 3.369
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Then: work out the number logs.
Last: check.
▶ Watch it solved
The question
Question: expand log₂(8√x).
Expand means: write it as single logs, added and taken away.
log2(8√x)
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 8 and √x multiplied: each gets its own log, with plus.
= log2 8 + log2 √x
2 Bring the powers down
A square root is the power one half: √x is x to the power 1/2.
√x = x½
Each power comes down in front, as times: 1/2 log₂ x.
= log2 8 + 12 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.
= 3 + 12 log2 x
23 = 8
4 Check
Let's check with numbers: x = 3.
With a calculator, the question is about 3.792.
x = 3: question ≈ 3.792
Our answer gives about 3.792 too.
It works!
answer ≈ 3.792
Let's look back.
First: split the times and divide.
Then: bring the powers down.
Then: work out the number logs.
Last: check.
▶ Watch it solved
