Math I Review
5. Indices and logarithms
5.1 Indices and logarithms
Index notation
In general, if
𝑀𝑀 = 𝑏𝑏 𝑛𝑛
We say that 𝑏𝑏 𝑛𝑛 is the exponential form of M to base b. The number n is then referred to as the
index, power o exponent.
When n=0 then
𝑏𝑏 0 = 1
Negative powers are evaluated by taking the reciprocal of the corresponding positive power
where n is any positive whole number.
𝑏𝑏 −𝑛𝑛 =
1
𝑏𝑏 𝑛𝑛
For any base b and fraction p/q, we define
𝑝𝑝
1
1
𝑏𝑏 𝑞𝑞 = (𝑏𝑏 𝑝𝑝 )𝑞𝑞 = (𝑏𝑏 𝑞𝑞 )𝑝𝑝
Rules of indices
There are four basic rules of indices which facilitate the manipulation of indices:
1.
2.
3.
4.
Logarithms
𝑏𝑏 𝑚𝑚 × 𝑏𝑏 𝑛𝑛 = 𝑏𝑏 𝑚𝑚+𝑛𝑛
𝑏𝑏 𝑚𝑚 ÷ 𝑏𝑏 𝑛𝑛 = 𝑏𝑏 𝑚𝑚−𝑛𝑛
(𝑏𝑏 𝑚𝑚 )𝑛𝑛 = 𝑏𝑏 𝑚𝑚𝑚𝑚
(𝑎𝑎𝑎𝑎)𝑛𝑛 = 𝑎𝑎𝑛𝑛 𝑏𝑏 𝑛𝑛
Quite generally,
𝑖𝑖𝑖𝑖 𝑀𝑀 = 𝑏𝑏 𝑛𝑛 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑙𝑙𝑙𝑙𝑙𝑙𝑏𝑏 𝑀𝑀 = 𝑛𝑛
Where n is called the logarithm of M to base b.
The rules of logarthms are as follows:
1. log 𝑏𝑏 (𝑥𝑥 × 𝑦𝑦) = log 𝑏𝑏 𝑥𝑥 + log 𝑏𝑏 𝑦𝑦
2. log 𝑏𝑏 (𝑥𝑥 ÷ 𝑦𝑦) = log 𝑏𝑏 𝑥𝑥 − log 𝑏𝑏 𝑦𝑦
3. log 𝑏𝑏 𝑥𝑥 𝑚𝑚 = 𝑚𝑚 log 𝑏𝑏 𝑥𝑥
Adapted from Mathematics for Economics and Business 9th edition – Ian Jacques – Pearson Education - 1
Math I Review
5.2 The exponential and natural logarithm functions
The function
is referred to as the exponential function.
𝑓𝑓(𝑥𝑥) = 𝑒𝑒 𝑥𝑥
For base e,
𝑖𝑖𝑖𝑖 𝑀𝑀 = 𝑒𝑒 𝑥𝑥 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝑛𝑛 = log 𝑒𝑒 𝑀𝑀 = ln 𝑀𝑀
We call logarithms to base e natural logarithms.
Adapted from Mathematics for Economics and Business 9th edition – Ian Jacques – Pearson Education - 2