CONNECT: Powers and logs 2
USES AND RULES OF LOGARITHMS
Logarithms are used extensively in situations where numbers are unwieldy –
either very large or very small. The main advantages of using logarithms
(shortened to logs) in these cases are that the numbers become more
comprehensible and calculations become much simpler.
One well-known use of logarithms is the Richter scale, used to measure the
strength of earthquakes. Another is pH, a measure of acidity. Decibels
(noise) is another example which uses logarithms as a unit of measure.
Logarithms are powers
Remembering that logarithms are powers, it follows that the rules of powers
also hold with logarithms.
p
q
The first of these rules is a x a = a
p+q
So to multiply two powers of the same base, we add the powers.
From this it can be shown that log 𝑎 ( 𝑚 × 𝑛) = log 𝑎 𝑚 + log 𝑎 𝑛
p
q
Another rule is a ÷ a = a
p-q
Again, it can be shown that log 𝑎 ( 𝑚 ÷ 𝑛) = log 𝑎 𝑚 − log 𝑎 𝑛
Further, a = 1 will give us that log 𝑎 1 = 0 and a = a will give us that
0
1
log 𝑎 𝑎 = 1
(𝑎𝑝 )𝑞 = 𝑎𝑝𝑞 will give us log 𝑎 𝑚𝑝 = 𝑝 log 𝑎 𝑚
Proofs of each of these relationships and exercises on their use can be found
in many high school Mathematics text books. All rely on the fact that
logarithms are powers. Proof of the first result follows overleaf.
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To show that log 𝑎 ( 𝑚 × 𝑛) = log 𝑎 𝑚 + log 𝑎 𝑛
Let log 𝑎 𝑚 = 𝑝 and log 𝑎 𝑛 = 𝑞
This means that 𝑎𝑝 = 𝑚 and 𝑎𝑞 = 𝑛
So, 𝑚 × 𝑛 = 𝑎𝑝 × 𝑎𝑞
= 𝑎𝑝+𝑞
Writing this in its log form: log 𝑎 (𝑚 × 𝑛) = 𝑝 + 𝑞
Replacing p and q with their original logs (from above) we see that
log 𝑎 ( 𝑚 × 𝑛) = log 𝑎 𝑚 + log 𝑎 𝑛
The other proofs are similar.
If you need help with any of the Maths covered in this resource (or any other
Maths topics), you can make an appointment with Learning Development
through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11,
or through your campus.
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