INDICIAL AND EXPONENTIAL EQUATIONS

CONTENT

• Exponential Equation of Linear Form
• Exponential Equation of Quadratic Form

Exponential Equation of Linear Form

Under exponential equations, 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   b)  (0.25) x+1  =  16    c)  3x = 1/81   d) 10 x = 1/0.001   e)  4/2x = 64 x

Solution 

a) (1/2) X = 8                                                     b) (0.25) x+1 = 16

(2 -1) x = 2 3                                                                (25/100) x+1 =  42                                    

  2 –x = 2 3                                                                 (1/4) x+1  =  42

  -x = 3                                                                      (4-1) x + 1  = 42           

   x = – 3                                                                     4 – x – 1   = 42

                                                                                                – x – 1 = 2

                                                                                    – x = 2 + 1   

                                                                                                 – x = 3

                                                                                                   X = – 3 

c) 3x = 1/81                                                               d) 10 x = 1/0.001

            3x = 1/34                                                                     10 x  = 1000

3x = 3 -4                                                                       10 x  = 10 3

x = -4                                                                         10 x  = 10 3

                                                                            x = 3

e) 4/2x = 64 x

4÷2x = 64 x

22 ÷2x = 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

Evaluation

Solve the following exponential equations

1. a) 2 x = 0.125 b) 25 (5x) = 625       c) 10 x = 1/100000

Exponential Equation of Quadratic Form

Some exponential equation can be reduced to quadratic form as can be seen below.

Example:  

Solve the following equations.

1. a) 22x – 6 (2x) + 8 = 0
2. b) 52x + 4 x 5 x+1 – 125 = 0
3. c)          32x – 9 = 0

Solution

1. a)   22x – 6 (2x) + 8 = 0                                 When y = 4 then,         and When y = 2 then,

(2x)2 – 6 (2x) + 8 = 0                                          2 x = 4                                       2 x = 2  

Let 2x = y                                                          2 x = 2 2                                                          2 x = 2 1

Then y2 – 6y + 8 = 0                                           x = 2                                         x = 1

Then factorize                                                                     x = 1 and 2

y 2 – 4y – 2y + 8 = 0

y (y – 4) -2 (y -4) = 0

(y -2) (y – 4) = 0

y – 2 = 0 or y – 4 = 0

y = 2 or y= 4

y = 2, 4

b) 52x + 4 x 5x+1 – 125 = 0

(5 x) 2 + 4 x (5 x x 51) – 125 = 0

Let 5 x = p

P 2 + 4 x (p x 5) – 125 = 0

P2 + 4 (5p) – 125 = 0

P2 + 20p – 125 = 0

Then, Factorize p2 + 25p – 5p – 125 = 0

                p (p + 25) – 5 (p + 25) = 0

                  (p – 5) (p + 25) = 0

                p – 5 = 0 p + 25 = 0

                p = 5 or p = – 25

                           Since 5x = p, p = 5

                                     5x  = 5 1

                                       x = 1

                           5x = -25 

c) 3 2x – 9 = 0

(3 x) 2 – 9 = 0

Let 3x  = a 

a2 – 9 = 0

a2 = 9

a = ±√9

a = ± 3

a = 3 or – 3

           Since 3x  = a, when a = 3

3 x  = 31

x = 1

            Since 3x = a, when a = -3

3 x = – 3

Evaluation:   

Solve: (a)   3(22x + 3) – 5(2x+2) – 156 = 0         (b ) 92x+1 = (81 x-2/3x)

General Evaluation

Solve the following exponential equations.

1. a) 22x + 1 – 5 (2x) + 2 = 0
2. b) 32x – 4 (3x+1) + 27 = 0

Reading Assignment: Further Mathematics Project Book 1(New third edition).Chapter 2 pg. 6- 10

Weekend Assignment

1. Solve for x : (0.25) X + 1 = 16                                       (a) -3        (b) 3        (c) 4       (d) -4
2. Solve for x : 3(3)X = 27                                               (a) 3         (b) 4        (c) 2       (d) 5
3. Solve the exponential equation : 22x + 2x+1 – 8 = 0   (a) 1         (b) 2        (c) 3       (d) 4
4. The second value of x in question 3 is                      (a) -1        (b) 1        (c) 2       (d)  No solution
5. Solve for x : 10 -X = 0.000001                                   (a) 4         (b) 6        (c) -6      (d) 5

Theory

Solve the following exponential equations

         (1)  (3x)2 + 2(3x)– 3 = 0          (2)  52x+1 – 26(5x) + 5 = 0