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Mathematics for science and technology
Mathematics for science and technology

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2 Three log rules

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

Rule one log sub a of x times y equals log of sub a of x plus log of sub a of y

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.

Rule two log sub a of x divided by y equals log of sub a of x minus log of sub a of y

Again, let logax = m and logay = n with x = am and y = an.

multiline equation row 1 x divided by y equals a super m divided by a super n row 2 equals a super open m minus n close

(from Week 1: Rule 2)

Applying the definition of a logarithm gives:

loga  x divided by y  = loga(a(m – n))

loga  x divided by y  = m – n

log sub a of x divided by y equals log of sub a of x minus log of sub a of y

For example, loga 7 = loga 14 – loga 2.

Rule three log sub a of postfix times x super r equals r log of sub a of postfix times x

Let logax = n with x = an

Raising each side to the power r gives

multiline equation row 1 multiline equation line 1 x super r equals left parenthesis a super n times right parenthesis super r row 2 equals multiline equation line 1 a super r times n

(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

Timing: Allow about 10 minutes
  1. Simplify  log 6 + log 3 – log 9
  2. Write  4 log x – ½ log y + 3 log z  as a single logarithm
  3. log 64 ÷ log 2 (Note it can be useful to consider if a number can be written in the 2n)
  4. (log 27 − log 9) ÷ log 3

Discussion

  1. 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)

  2. Using Rule 3: 4 log x – ½ log y + 3 log z = log x4 – log y½ + log z3

    Then using Rule 2: log of x super four divided by y super one divided by two plus log of postfix times z cubed

    The Rule 3: log of open x super four times z cubed divided by y super one divided by two close

  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

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

    multiline equation row 1 log of open 27 division nine close division log of open three close equals log of prefix of of open three close divided by log of open three close row 2 equals one

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