## Logs

A log has a base and a number: e.g. log38

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:

log5125 means: “what power do I need to raise 5 to, to get 125?

∴ log5125=3 (because 53 = 125)

log264 means: “what power do I need to raise 2 to, to get 64?

∴ log264=6 (because 26 = 64)

log100 means: “what power do I need to raise 10 to, to get 100?

∴ log100=2 (because 102 = 100)

### Change of base

Most new calculators will accept a log to any base by using the general log key ( )

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:

Example:

·  Find log435 using the log to the base 10 key.

log435=log1035log104=2.56

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 and ln “undo” each other because they are “opposites”.

e.g. lne3=3 (because the ln and the e undo each other (or cancel each other out))

eln5=5 (because the ln and the e undo each other (or cancel each other out))

### Laws of Logs

There are a few rules/laws related to logs that it is useful to learn and know how to use.

logAB=logA+log(B)

logAB=logA-log⁡(B)

logAn=nlog(A)

log1=0 (using ANY base)

#### Examples:

·  Write log4+log⁡(5) as a single log.

Using the 1st rule: log4+log5=log4×5=log⁡(20)

·  Write log12-log⁡(3) as a single log.

Using the 2nd rule: log12-log3=log123=log⁡(4)

·  Find log151.

Using the 4th rule: log151=0

·  Write 2log45+3log4⁡(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: 2log45+3log47= log452+log473

Using the 1st rule: log452+log473=log452×73=log4⁡(8575)

·  Given that log5≈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: log125=log5×5×5 =log5+log5+log5

=0.699+0.699+0.699=2.097

OR:

Using the 3rd rule: log125=log53 =3log⁡(5)

=3×0.699=2.097

© H Jackson 2011 / ACADEMIC SKILLS 3