# 2 Three log rules

The three log rules can be deduced from the rules of indices that you learned in Week 1.

If you let log_{a} *x* = *m* and log_{a} *y = n*

This means that *x* = *a*^{m}and *y* = *a*^{n}

So *xy* = *a** ^{m}* ×

*a*

*=*

^{n}*a*

^{(}

^{m + n}^{)}(from Week 1: Rule 1).

Applying the definition of a logarithm gives:

log_{a} *xy* = log_{a} a^{(}^{m + n}^{)}

log_{a} *xy* = *m* + *n*

log_{a} *xy* = log_{a} *x* + log_{a} *y*

For example, log* _{a}* 21 = log

*7 + log*

_{a}*3, since 7 and 3 are both factors of 21.*

_{a}Again, let log_{a}*x* = *m* and log_{a}*y* = *n* with *x* = *a** ^{m}* and

*y*=

*a*

*.*

^{n}(from Week 1: Rule 2)

Applying the definition of a logarithm gives:

log_{a} = log_{a}(*a*^{(m – n)})

log_{a} = *m* – *n*

For example, log* _{a}* 7 = log

*14 – log*

_{a}*2.*

_{a}Let log_{a}*x* = *n* with *x* = *a*^{n}

Raising each side to the power *r* gives

(from Week 1: Rule 3)

Applying the definition of a logarithm gives:

log_{a}*x** ^{r}* = log

_{a}(

*a*

*)*

^{rn}For example, log_{10} 1000 = log_{10} 10^{3} = 3 log_{10} 10 = 3 (because log_{10} 10 = 1)

Use the rules in this next activity.

## Activity 3 Using the three log rules

- Simplify log 6 + log 3 – log 9
- Write 4 log
*x*– ½ log*y*+ 3 log*z*as a single logarithm - log 64 ÷ log 2 (Note it can be useful to consider if a number can be written in the 2
^{n}) - (log 27 − log 9) ÷ log 3

### Discussion

Using Rule 1: log (6) + log (3) = log (6 × 3)

Giving: log (18) – log (9)

Using Rule 2: log (18) – log (9) = log (2)

So, log (6) + log (3) = log (2)

Using Rule 3: 4 log

*x*– ½ log*y*+ 3 log*z*= log*x*^{4}– log*y*^{½}+ log*z*^{3}Then using Rule 2:

The Rule 3:64 = 2

^{6}log (64) ÷ log (2) = log (2

^{6}) ÷ log (2)Using rule 3: log (2

^{6}) = 6 log (2)log (2

^{6}) ÷ log (2) = 6 log (2) ÷ log (2)= 6

Using Rule 2: log (27) – log (9) = log (27 ÷ 9)

In the next section you will learn about a special logarithm, called the natural logarithm.