1. Chapter 7 – Indices and Logarithms07
Chapter 7 – Indices and Logarithms
Learning Outcomes
In this chapter you have learned about:
Exponents or indices
The exponential function
The natural exponential function
The laws of indices
The methods for solving equations of the form 𝑎 𝑥 = b, where a and b are constants and 𝑥 𝜖 𝑅
Surds
Logarithms
The laws of logarithms
Using logs to solve real-life problems

2. Indices and Logarithms07
Indices and Logarithms
Indices (Exponents) and Exponential Functions
Definitions
A number in index form is of the form 𝑏 𝑛 . We call b the base and n the index, power or exponent.
Index notation is useful for writing very large or small numbers.
For example, 10,000,000,000,000,000 can be written as 1016
Exponential Functions
Graphs of four exponential functions.
𝒇 𝒙 = 𝟑 −𝒙
𝒈 𝒙 = 𝟑 𝒙
All graphs of will pass through
the point (0,1) as 𝑎 0 =1. -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 x y
𝒇 𝒙 = 𝟐 𝒙
𝒇 𝒙 = 𝟐 −𝒙
Natural Exponential Function -6 =5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 x y
𝑒=1+ 1 1! + 1 2! + 1 3! + 1 4! …… =
𝒇 𝒙 = 𝒆 𝒙
e is an irrational number

3. Indices and Logarithms07
Indices and Logarithms
Laws of Indices (Exponents)
(All in F and T p. 21)
Law 1: 𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞
Example: 𝑥 2 × 𝑥 3 = 𝑥 5
Law 2: 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞
Example: 𝟑 𝟓 𝟑 𝟐 = 𝟑 𝟑 =𝟐𝟕
Law 3: ( 𝒂 𝒑 ) 𝑞 = 𝑎 𝑝𝑞
Example: ( 𝟓 𝟐 ) 𝟒 = 𝟓 𝟖
Law 4: 𝑎 0 =1
Example: 𝟔 0 =1
Example: 𝒚 1 𝟑 = 𝟑 𝒚
Law 5: 𝑎 −𝑝 = 1 𝑎 𝑝
Law 6: 𝑎 1 𝑞 = 𝑞 𝑎
Example: 𝟒 𝟑 𝟐 =( 𝟐 𝟒 ) 𝟑 =𝟖
Law 7: 𝑎 𝑝 𝑞 =( 𝑞 𝑎 ) 𝑝
Example: 𝟓 −𝟐 = 1 𝟓 𝟐 = 𝟏 𝟐𝟓
Law 8: (ab ) 𝑝 = 𝑎 𝑝 𝑏 𝑝
Example: (𝒙𝒚 ) 𝟑 = 𝒙 𝟑 𝒚 𝟑
Law 9: 𝑎 𝑏 𝑝 = 𝑎 𝑝 𝑏 𝑝
Example: 𝟑 𝟐 𝟒 = 𝟑 𝟒 𝟐 𝟒 = 𝟖𝟏 𝟏𝟔

4. Indices and Logarithms07
Indices and Logarithms
Equations with x as an Index
Solve 𝟒 𝒙 = 𝟖 𝟐
All numbers in the equation can be written as powers of 2.
( 2 2 ) 𝑥 =
2 2𝑥 = 2 2.5
2𝑥=2.5
𝑥=1.25
Solve the equation 𝟑 𝟐𝒙+𝟏 −𝟐𝟖 𝟑 𝒙 +𝟗=𝟎
3 2𝑥+1 −28 3 𝑥 +9=0
( 3 2 ) 𝑥 (3 ) 1 −28 3 𝑥 +9=0
As 𝑦 =3 𝑥
3 𝑥 = 𝐨𝐫 𝑥 =9
3( 3 𝑥 ) 2 −28 3 𝑥 +9=0
Let 𝑥 =𝑦
3 𝑥 = 3 − 𝐨𝐫 𝑥 = 3 2
3 𝑦 2 −28𝑦+9=0
⇒𝑥=− 𝐨𝐫 𝑥=2
(3𝑦−1)(𝑦−9)=0
𝑦= 𝐨𝐫 𝑦=9

5. Indices and Logarithms07
Indices and Logarithms
Surds
Definitions
Roots that are irrational are called surds.
Law 1: 𝑎 𝑏 = 𝑎𝑏
Example: 𝟐𝟎 = (𝟒)(𝟓) = 𝟒 × 𝟓 =𝟐 𝟓
Law 2: 𝒂 𝒃 = 𝒂 𝒃
Example: 𝟓 𝟗 = 𝟓 𝟗 = 𝟓 𝟑
Rationalise the denominator of:
𝟓 𝟓
= = 5
Rationalise the denominator of:
𝟐 − 𝟑 𝟐 + 𝟑
( 2 − 3 )( 2 − 3 ) ( )( 2 − 3 )
5−2 6 −1
2− 6 − − −3
−5+2 6

6. Indices and Logarithms07
Indices and Logarithms
Logarithms
Index form:
23 = 8
Log form:
log2 8 = 3
Definitions
The graph function
f(x) = log2x, x ∈ R, x > 0 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 -4 -3 -2 x y
f(x) = log2x
The graph of y = log2 x is a reflection of the graph of y = 2x in the line y = x. -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 x y
y = 2x
y = x
y = log2x

7. Indices and Logarithms07
Indices and Logarithms
The Laws of Logarithms
(All in F and T p. 21)
Law 2: 𝑙𝑜𝑔𝑎 𝒙 𝒚 = 𝑙𝑜𝑔 𝑎 𝑥− 𝑙𝑜𝑔 𝑎 𝑦
Law 1: 𝑙𝑜𝑔𝑎 𝑥𝑦 = 𝑙𝑜𝑔 𝑎 𝑥+ 𝑙𝑜𝑔 𝑎 𝑦
Law 5: 𝑙𝑜𝑔𝑎 1 𝑥 = −𝑙𝑜𝑔 𝑎 𝑥
Law 3: 𝑙𝑜𝑔𝑎 𝑥 𝑞 =𝑞 𝑙𝑜𝑔 𝑎 𝑥
Law 4: 𝑙𝑜𝑔𝑎 1=0
Law 8: 𝑙𝑜𝑔𝑏 𝑥= 𝑙𝑜𝑔 𝑎 𝑥 𝑙𝑜𝑔 𝑎 𝑏
Law 6: 𝑙𝑜𝑔𝑎 𝑎 𝑥 =𝑥
Law 7: 𝑎 𝑙𝑜𝑔 𝑎 𝑥 =𝑎
= 1 𝑙𝑜𝑔 2 𝑥 + 3
4 1 𝑙𝑜𝑔 2 𝑥
If log𝟏𝟎(𝟑𝒙+𝟏)=𝟐, find 𝒙.
log10 (3𝑥 + 1) = 2
⇒4 1 𝑦 =𝑦+3
Let 𝑦=𝑙𝑜𝑔2 𝑥
⇒10 2 =3𝑥+1
⇒𝑦 2 +3𝑦−4=0
3𝑥+1=100
𝑦+4 𝑦−1 =0
3𝑥=99
𝑥=33
⇒𝑦=−4 𝐨𝐫 𝑦=1
Ensure that all logs have the same base.
𝑙𝑜𝑔2 𝑥=−4 𝐨𝐫 𝑙𝑜𝑔2 𝑥=1
Solve for 𝒙 if 𝟒𝒍𝒐𝒈𝒙 𝟐=log𝟐 𝒙+𝟑, 𝒙>𝟏, 𝒙∈𝑹
𝑥= 2 − 𝐨𝐫 𝑥= 2 1
𝑙𝑜𝑔𝑥 2= 𝑙𝑜𝑔 2 2 𝑙𝑜𝑔 2 𝑥
= 1 𝑙𝑜𝑔 2 𝑥
Law 8
𝑥= 𝐨𝐫 𝑥=2
As 𝑥 > 1, the solution is 𝑥=2

8. Indices and Logarithms07
Indices and Logarithms
Using Logarithms to Solve Practical Problems
Many mathematical models that represent real life situations contain unknown powers or indices.
𝐹=𝑃(1+𝑖)𝑡 Compound interest formula
𝑝=𝑎𝑒𝑏𝑡 Exponential model: describes population growth, radioactive decay, etc
If b > 0, then p is growing. If b < 0, then p is decaying.
𝑀= 𝑙𝑜𝑔 10 𝐼 𝑆
Richter scale
𝐷= 10𝑙𝑜𝑔 10 𝐼 𝐼 0
Decibel scale: measures loudness of sound as perceived by the human brain.
𝑝𝐻= −𝑙𝑜𝑔 10 𝐻 +
pH scale:  measures how acidic or basic a substance is.