Indices And Logarithm SS1 Further Mathematics Lesson Note

Basic Concept of Laws of Indices

A number of the form am where a is a real number, a is multiplied by itself m times,

The number a is called the base and the superscript m is called the index (plural indices) or exponent. 

1. am x  an =  a(m + n) = Multiplication law

Example:  p3   x  p2  = ( p  x  p x  p)  x  (p  x  p)  =  p 5 

Or p3   x  p2 =  p 3 + 2 =  p5

2. am ÷ an   = a(m – n) = Division law

Example:  p6  ÷  p4  =  p 6  –  4 = p2

3. (am)n = a(mn) = Power law

Example:  (p3)2 = p3 x p3 =  p 3 + 3 =  p6 

Or  p3 x 2 = p6

4. am ÷ am  =  a(m – m)  =  a0  = 1 

 am ÷am = am/am = ao = 1

 a0 = 1  Zero Index

Note: Any number raised to power of  zero is 1

Example:  3o = 1,   co = 1,    yo = 1

5. (ab)m = amb = Product power law

 e.g. (2xy)2  = 4x2y2   

6. a^-m = 1/a^m = Negative Index

Example:  2 -1 =  ½,    and   3 -2  =  1/3 2 = 1/9

7. a1/n  = n√a = Root power law

Example :   9 ½ = √9 = 3

/3 =3√27 = 3 ie (3)3 = 3

Exponential Equation of Linear Form

Under an exponential equation, if the base numbers of any equation are equal, then the power will be equal & vice versa.

Examples:

Solve the following exponential equations

1. a)  (1/2) x  =  8   
2. b)  (0.25) x+1  =  16   
3. c)  3x = 1/81
4. d) 10 x = 1/0.001   
5. e)  4/2x = 64 x

Solution 

1. a) (1/2) X = 8      

(2 ^ -1) x = 2 ^ 3 

2 ^x = 2 ^3

-x = 3

x = – 3

1. b) (0.25) x+1 = 16

(25/100) x+1 =  4^2                                                                                                    (1/4) ^(x+1)  =  4^2

(4-1)^ (x + 1)  = 4^2           

4 ^ (x – 1)   = 4^2                                                                        – x – 1 =2                                                                               – x = 2 + 1                                                                                  – x = 3                                                                                    X = – 3 

1. c) 3x = 1/81          

3x = ⅓^4                                                                     3x = 3^ -4                                                                        x = -4  

1. d) 10 x  = 1/0.001                                                                     10 ^ x  = 1000                                                                      10^x  = 10^3

10 ^ x  = 10 ^ 3                                                                            x = 3

1. e) 4/2^x = 64 ^ x

4÷2^x = 64^ x

2^2 ÷2^x = 64 ^ x

2^2-x = (2 ^ 6)^x

2 ^ 2-x = 2^ 6x

2- ^x = 6x

2=6x+x

2 = 7x

Divide both sides by 7

2/7 = 7x/7

x = 2/7

Logarithm of numbers (Index & Logarithmic Form)

The logarithm to base a of a number P is the index x to which a must be raised to be equal to P.

Thus if P = axe, then x is the logarithm to the base a of P. We write this as x = log a P. The relationship logaP = x and 

axe =P is equivalent to each other.

ax =P is called the index form and logaP = x is called the logarithm form

Conversion from Index to Logarithmic Form

Write each of the following in index form in their logarithmic form

1. a) 2^6 = 64
2. b) 25^½ = 5
3. c) 4^4= 1/256

Solution

1. a) 2^6 = 64

Log2 64 = 6

1. b) 25^½ = 5

Log25(5)=1/2

1. c) 4^-4= 1/256

Log4(1/256 ) = -4

Conversion from Logarithmic to Index form

1. a) Log2128 = 7
2. b) log10 (0.01) = -2                  
3. c) Log1.5 2.25 = 2

Solution

1. a) Log2(128) = 7

2^7 = 128

1. b) Log10 (0.01) = -2

10^-2= 0.01

1. c) 5 (2.25) = 2

1.5^2 = 2.25

Laws of Logarithm

1. a) let P = bx, then logbP = x

Q = by, then logbQ = y 

PQ = bx X by = bx+y (laws of indices)

Logb PQ = x + y

:. Logb PQ = logbP + LogbQ

1. b) P÷Q = bx÷by = bx+y

LogbP/Q = x y

:. LogbP/Q = logbP  logbQ

1. c) Pn= (bx)n = bxn

 Logbpn = nbx

:. LogPn = logbP

1. d) b = b1 

:.  Logbb = 1

1. e) 1 = b0

Logb1 = 0

Example  

Solve each of the following:

1. a) Log327 + 2log39  log354
2. b) Log313.5  log310.5
3. c) Log28 + log23
4. d) Given that log102 = 0.3010 log103 = 0.4771 and log105 = 0.699 find the log1064 + log1027

Solution

1. Log327 + 2 log39  log354

= log3 27 + log3 92 log354

= log3 (27 x 92/54) 

= log3 (271 x 81/54) = log3 (81/2)

= log3 34/ log32

= 4log3 3  log3 2

= 4 x (1)  log3 2 = 4  log3 2 

= 4 – log3 2

1. log3 13.5 – log3 10.5

= log3 (13.5) – Log310.5 = log3 (135/105) 

= log3 (27/21) = log3 27 – log3 21

= log3 33 – log3 (3 x 7)

= 3log3 3 – log3 3 -log37

= 2 – Log3 7

1. Log28 + Log33

= log223+ log33 

= 3log22 + log33

= 3 +1 = 4

1. log10 64 + log10 27

log10 26 + log1033

6 log10 2 + 3 log 10^3

(0.3010) + 3(0.4771)

1.806 + 1.4314 = 3.2373.