## 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