Indices, Surds &

Logarithms

Indices

54= 5 x 5 x 5 x 5 = 625

5 is the base

4 is the index or power

Laws of Indices

am ´ an = am+n

am ¸ an = am-n

(am )n = amn

am ´bm = (a´b)m

am ¸ bm = (a

b)m

a0 =1

a-m =1

am

(a

b)-m = (

b

a)m

a1

m = am

an

m = ( am )n = a nm

(am ´ bn )l = aml ´bnlGotta Memorize these!!!

Example Questions

3x+y ´ 32 x-y

= 3(x+y)+(2 x-y)

= 33x

6x+y ¸ 6x-y

= 6(x+y)-(x-y)

= 62 y

Q1 Q2

Example Questions

33 ´ x3

= (3´ x)3

= (3x)3

x6 ¸ 26 = (x

2)6

(215)x = 215x

3=7 =1

37

(16

7)-3 = (

7

16)3

Q3

Q4

Q5

Q6

Q7

Example Questions

1. Evaluate (18)3

2 ´ (6)-

1

2 ´1

27

= (2 ´32 )3

2 ´ (2 ´3)-

1

2 ´1

(33)1

2

= (2 ´32 )3

2 ´ (2 ´3)-

1

2 ´ (33)-

1

2

= 23

2 ´33 ´ 2-

1

2 ´3-

1

2 ´ 3-

3

2

= 23

2-

1

2 ´33-

1

2-

3

2

= 21 ´31

= 6

2. Solve the equation 81n = 9

81= 34

9 = 32

Hence, equation is (34 )n = 32

34n = 32

4n = 2

n =1

2

Example Questions

3. Simplify 3n+1 ´3n-1

9-n

=3(n+1)+(n-1)

3-2n

=3n+1+n-1

3-2n

= 32n-(-2n)

= 34n

4. Solve the equation 22 x+1 - 3(2x )+1= 0

(22 x ´ 21)-3(2x )+1= 0

Let 2x be y

2y2 - 3y+1= 0

(2y-1)(y-1) = 0

y =1

2, y =1

2x =1

2® 2x = 2-1 ® x = -1

2x =1® 2x = 20 ® x = 0

Example Questions

5. Solve the equation 9x ´ 22 x = 6

32 x ´ 22 x = 6

(3´ 2)2 x = 6

62 x = 61

2x =1

x =1

2

6. Solve the equation 81x = 273x-5

34 = 33(3x-5)

34 = 39 x-15

4 = 9x -15

9x =15+ 4

9x =19

x = 21

9

Surds surds is a subset of irrational numbers

Eg √2, √3, √6 etc

Multiplication of surds

√a × √b = √ab

Eg √5 × √7 = √30

Addition and Subtraction of surds

Eg 3√2 + 2√2 − 4√2 = 5√2 – 4√ 2 = √2

Division of surds

Eg √100 ÷ √25 = √(100÷25) = √4 = 2

Rationalizing Surds

Q2.2 +1

11 +3=

( 2 +1)

( 11 +3)´

( 11 -3)

( 11 -3)

=( 2 +1)( 11 -3)

11- 9

=( 2 +1)( 11 -3)

2

Q1.2

3=

2

3´

3

3

=2 3

3

Rationalize if there are

surds in the denominator

(a+b)(a-b) = a2 – b2

Example Questions

1. Simplify.

1

5+ 20 + 125

=1

5*

5

5+ 2 ´ 2 ´ 5 + 5´ 5´ 5

=5

5+ 2 5 + 5 5

= 5(1

5+ 2 + 5)

= 71

55

2. Simplify and express in the from a+b c

( 5 - 2)2

= ( 5)2 - 2( 5)(2)+ 22

= 5- 4 5 + 4

= 9 - 4 5

Laws of Logarithm If y=a x, x is defined as the logarithm of y to the base a.

® x = loga y

1. loga xy = loga x + loga y

2. logax

y= loga x - log a y

3. log(x)n = n loga x

4. loga xn = loga x1

n =1

nloga x

Note:

- The log of any negative number to any base does not exist

eg. log5(-10) does not exist

- The log of 1 to any base is zero

eg. log31= 0

- logxx =1

Logarithm Logarithms to base 10 are called common logarithms

Change of base

log10 a® loga or lga

loga x =logb x

logb a

loga x =1

logx a

loga b =logb

loga

How to calculate?

Example Questions

1. Solve the equation 2 x ´3x = 5x+1

2x ´3x = 5x+1

6x = 5x+1

x log10 6 = (x +1)log10 5

x +1

x=

log10 6

log10 5

=0.7782

0.6990

=1.113

x +1=1.113x

(1.113-1)x =1

x = 8.85 (2d.p)

2. Simplify log3 2 + log3 5+ log3 20 - log3 25

= log3(2 ´ 5´ 20

25)

= log3(200

25)

= log3 8

= log3 23

= 3log3 2

Example Questions 3. Solve the equation

(log5 x)2 - 3log5 x + 2 = 0

Let log5 x = y

y2 - 3y+ 2 = 0

(y- 2)(y-1) = 0

y = 2, y =1

log5 x = 2

x = 52 = 25

log5 x =1

x = 51 = 5

4. Solve the equation log4 x - logx 8 =1

2

log4 x -log4 8

log4 x=

1

2

log4 x -1.5

log4 x=

1

2

let log4x be y

y-1.5

y=

1

2

´ y : y2 -1.5 =1

2y

y2 -1

2y-1.5 = 0

´ 2: 2y2 - y- 3 = 0

(2y-3)(y+1) = 0

2y = 3, y = -1

y =3

2

log4 x =3

2

x = 43

2

x = 8

y = -1

log4 x = -1

x = 4-1

x =1

4

Example Questions

5. Solve the equation

log9[log2 (4x -16)] = log16 4

log9[log2 (4x -16)] =1

2

log9[log2 (4x -16)] = log3 9

log2 (4x -16) = 3

log2 (4x -16) = log2 8

4x -16 = 8

4x = 24

x = 6

6. Solve the simultaneous equations

log2 xy = 2® eqn1

logx 4y = 6® eqn2

eqn1: y = (2x)2

y = 4x2 ® eqn3

sub eqn3 into eqn2

logx 4(4x2 ) = 6

logx16x2 = 6

logx16x2 = logx x6

16x2 = x6

16x2 - x6 = 0

x2 (16 - x4 ) = 0

x2 = 0(rejected),16 - x4 = 0

16 = x4

x = 2,-2(rejected)

Exponential Function General form is ax where a is a positive constant and x is

a variable

Important exponential functions

10x

ex

Example Questions

1. Solve the equation

e2 ln x + lne2 x = 8

e2 ln x + 2x = 8

e2 ln x = 8- 2x

2 ln x = ln(8- 2x)

ln x2 = ln(8- 2x)

x2 = 8- 2x

x2 -8+ 2x = 0

(x + 4)(x - 2) = 0

x = -4(rejected), x = 2

2. Solve the equation

10x = e2 x+1

ln10x = lne2 x+!

x ln10 = 2x +1

x ln10 - 2x =1

x(ln10 - 2) =1

x =1

ln10 - 2

x = 3.30

Example Questions 3. Solve the equation

2e2 x+! = ex+1 +15e

2e2 x *e = ex *e+15e

let ex be y

2ey2 = ey+15e

2ey2 - ey-15e = 0

e(2y2 - y-15) = 0

2y2 - y-15 = 0

(2y+ 5)(y- 3) = 0

y = -5

2, y = 3

ex = -5

2(rejected)

ex = 3

x =1.10

4. Solve the equation

ex = 2ex

2 +15

ex = 2(ex )1

2 +15

let ex be y

y=2y1

2 +15

y- 2y1

2 -15 = 0

(y1

2 - 5)(y1

2 + 3) = 0

y1

2 = 5, y1

2 = -3(rejected)

y = 25

ex = 25

x = ln25 = 3.22

5. Solve the equation

e3x-1 =148

lne3x-1 = ln148

3x -1= 5

3x = 6

x = 2