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 loga x = m and loga y = n
This means that x = amand y = an
So xy = am × an = a(m + n) (from Week 1: Rule 1).
Applying the definition of a logarithm gives:
loga xy = loga a(m + n)
loga xy = m + n
loga xy = loga x + loga y
For example, loga 21 = loga 7 + loga 3, since 7 and 3 are both factors of 21.
Again, let logax = m and logay = n with x = am and y = an.
(from Week 1: Rule 2)
Applying the definition of a logarithm gives:
loga = loga(a(m – n))
loga = m – n
For example, loga 7 = loga 14 – loga 2.
Let logax = n with x = an
Raising each side to the power r gives
(from Week 1: Rule 3)
Applying the definition of a logarithm gives:
logaxr = loga(arn)
For example, log10 1000 = log10 103 = 3 log10 10 = 3 (because log10 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 2n)
- (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 x4 – log y½ + log z3
Then using Rule 2:
The Rule 3:64 = 26
log (64) ÷ log (2) = log (26) ÷ log (2)
Using rule 3: log (26) = 6 log (2)
log (26) ÷ 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.