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^mand y = aⁿ

So xy = a^m × aⁿ = 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_a 7 + log_a 3, since 7 and 3 are both factors of 21.

Again, let log_ax = m and log_ay = n with x = a^m and y = aⁿ.

(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_a 14 – log_a 2.

Let log_ax = n with x = aⁿ

Raising each side to the power r gives

(from Week 1: Rule 3)

Applying the definition of a logarithm gives:

log_ax^r = log_a(a^rn)

For example, log₁₀ 1000 = log₁₀ 10³ = 3 log₁₀ 10 = 3 (because log₁₀ 10 = 1)

Use the rules in this next activity.

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