Academic Skills Advice
Logs
A log has a base and a number: e.g.
𝑙𝑜𝑔3 8
base number
This would be read as: “log to the base 3 of 8”.
The most common base is base 10. If you see a log without the base number (e.g. on
your calculator) then you should assume it is base 10.
e.g. 𝑙𝑜𝑔45
is usually read as “log 45” and means “log to the base 10 of 45”.
What does a log mean?
Examples:
𝒍𝒐𝒈𝟓 𝟏𝟐𝟓 means: “what power do I need to raise 5 to, to get 125?”
∴ 𝑙𝑜𝑔5 125
=3
(because 53 = 125)
The answer to the log
𝒍𝒐𝒈𝟐 𝟔𝟒 means:
∴ 𝑙𝑜𝑔2 64
“what power do I need to raise 2 to, to get 64?”
=6
𝒍𝒐𝒈𝟏𝟎𝟎 means:
∴ 𝑙𝑜𝑔100
© H Jackson 2011 / ACADEMIC SKILLS
(because 26 = 64)
“what power do I need to raise 10 to, to get 100?”
=2
(because 102 = 100)
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Change of base
Most new calculators will accept a log to any base by using the general log key (
Enter base here
)
Enter number
here
If you have an older calculator (without the general log key) and need to find a log with a
different base then you can convert to base 10 using the formula:
𝑙𝑜𝑔𝑥 𝐴 =
𝑙𝑜𝑔10 𝐴
𝑙𝑜𝑔10 𝑥
Remember to do: log of number
divided by log of base.
Example:

Find 𝒍𝒐𝒈𝟒 𝟑𝟓 using the log to the base 10 key.
𝑙𝑜𝑔4 35 =
𝑙𝑜𝑔10 35
𝑙𝑜𝑔10 4
= 𝟐. 𝟓𝟔
On your calculator you just need
to type: 𝒍𝒐𝒈(𝟑𝟓) ÷ 𝒍𝒐𝒈(𝟒)
Try finding the logs from the previous examples using your calculator (and either the
general log key or the change of base formula).
What about ln?
Ln is a special log because it’s base is e (e is a number approximately equal to 2.718)
Example:
Ln 7 means “what power do I need to raise ‘e’ to, to get 7?”
Ln 7 = 1.946 (3dp) because e1.946 = 7
Remember:
e.g.
e and ln “undo” each other because they are “opposites”.
ln(𝑒 3 ) = 3
𝑒 𝑙𝑛5 = 5
(because the 𝑙𝑛 and the 𝑒 undo each other (or cancel each other out))
(because the 𝑙𝑛 and the 𝑒 undo each other (or cancel each other out))
© H Jackson 2011 / ACADEMIC SKILLS
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Laws of Logs
There are a few rules/laws related to logs that it is useful to learn and know how to use.
𝒍𝒐𝒈(𝑨𝑩) = 𝒍𝒐𝒈(𝑨) + 𝒍𝒐𝒈(𝑩)
𝑨
𝒍𝒐𝒈 ( ) = 𝐥𝐨𝐠(𝑨) − 𝐥𝐨𝐠(𝑩)
𝑩
𝒍𝒐𝒈(𝑨𝒏 ) = 𝒏𝒍𝒐𝒈(𝑨)
𝒍𝒐𝒈(𝟏) = 𝟎 (using ANY base)
Examples:

Write 𝐥𝐨𝐠(𝟒) + 𝐥𝐨𝐠(𝟓) as a single log.
Using the 1st rule:

Write 𝐥𝐨𝐠(𝟏𝟐) − 𝐥𝐨𝐠(𝟑) as a single log.
Using the 2nd rule:

12
log(12) − log(3) = log ( 3 ) = log(4)
Find 𝒍𝒐𝒈𝟏𝟓 𝟏.
Using the 4th rule:

log(4) + log(5) = log(4 × 5) = log(20)
𝑙𝑜𝑔15 1 = 0
Write 𝟐 𝐥𝐨𝐠 𝟒 (𝟓) + 𝟑𝐥𝐨𝐠 𝟒 (𝟕) as a single log.
We can’t just multiply the 5 and the 7 because of the numbers in front of the logs. We
need to deal with them 1st.

Using the 3rd rule:
2 log 4 (5) + 3 log 4 (7) = log 4 (52 ) + log 4 (73 )
Using the 1st rule:
log 4 (52 ) + log 4 (73 ) = log 4 (52 × 73 ) = log 4 (8575)
Given that 𝐥𝐨𝐠(𝟓) ≈ 𝟎. 𝟔𝟗𝟗, find 𝒍𝒐𝒈(𝟏𝟐𝟓) without using a calculator.
You could use the 1st OR the 3rd rule for this question.
Using the 1st rule:
OR:
Using the 3rd rule:
© H Jackson 2011 / ACADEMIC SKILLS
log(125) = log(5 × 5 × 5) = log(5) + log(5) + log(5)
= 0.699 + 0.699 + 0.699 = 𝟐. 𝟎𝟗𝟕
log(125) = log(53 )
= 3log(5)
= 3 × 0.699 = 𝟐. 𝟎𝟗𝟕
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