Chapter 7 – Indices and Logarithms 07 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

Indices and Logarithms 07 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! …… =2.71828182845904523536 𝒇 𝒙 = 𝒆 𝒙 e is an irrational number

Indices and Logarithms 07 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

Indices and Logarithms 07 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 3 2 1 2 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 = 1 3 𝐨𝐫 3 𝑥 =9 3( 3 𝑥 ) 2 −28 3 𝑥 +9=0 Let 3 𝑥 =𝑦 3 𝑥 = 3 −1 𝐨𝐫 3 𝑥 = 3 2 3 𝑦 2 −28𝑦+9=0 ⇒𝑥=−1 𝐨𝐫 𝑥=2 (3𝑦−1)(𝑦−9)=0 𝑦= 1 3 𝐨𝐫 𝑦=9

Indices and Logarithms 07 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 5 5 5 = 5 5 5 = 5 Rationalise the denominator of: 𝟐 − 𝟑 𝟐 + 𝟑 ( 2 − 3 )( 2 − 3 ) ( 2 + 3 )( 2 − 3 ) 5−2 6 −1 2− 6 − 6 +3 2− 6 + 6 −3 −5+2 6

Indices and Logarithms 07 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

Indices and Logarithms 07 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 −4 𝐨𝐫 𝑥= 2 1 𝑙𝑜𝑔𝑥 2= 𝑙𝑜𝑔 2 2 𝑙𝑜𝑔 2 𝑥 = 1 𝑙𝑜𝑔 2 𝑥 Law 8 𝑥= 1 16 𝐨𝐫 𝑥=2 As 𝑥 > 1, the solution is 𝑥=2

Indices and Logarithms 07 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.