INDICIAL AND EXPONENTIAL EQUATIONS

FURTHER MATHEMATICS SSS1 FIRST TERM

WEEK 2

INDICIAL AND EXPONENTIAL EQUATIONS

Contents

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

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

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

            3^x = 1/3⁴                                                                     10 ^x  = 1000

          3^x = 3 ^-4                                                                       10 ^x  = 10 ³

           x = -4                                                                         10 ^x  = 10 ³

                                                                                      x = 3

e)       4/2^x = 64 ^x

          4÷2^x = 64 ^x

          2² ÷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

 

Evaluation

Solve the following exponential equations

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

 

Exponential Equation of Quadratic Form

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

Example: 

Solve the following equations.

a)       2^2x – 6 (2^x) + 8 = 0

b)       5^2x + 4 x 5 ^x+1 – 125 = 0

c)          3^2x – 9 = 0

 

Solution

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

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

Let 2^x = 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)       5^2x + 4 x 5^x+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 5^x = p,   p = 5

                                     5^x  = 5 ¹

                                       x = 1

                           5^x = -25

                     

c)       3 ^2x – 9 = 0

          (3 ^x) ² - 9 = 0

          Let 3^x  = a

           a² – 9 = 0

          a² = 9

          a = ±√9

          a = ± 3

          a = 3 or – 3

           Since 3^x  = a,        when a = 3

          3 ^x  = 3¹

          x = 1

            Since 3^x = a,        when a = -3

          3 ^x = - 3

         

      

Evaluation:  

Solve: (a)   3(2^{2x + 3}) - 5(2^x+2) - 156 = 0         (b )      9^2x+1 = (81 ^x-2/3^x)

   

General Evaluation

Solve the following exponential equations.

a)       2^{2x + 1} – 5 (2^x) + 2 = 0

b)       3^2x – 4 (3^x+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: 2^2x + 2^x+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)  (3^x)² + 2(3^x)– 3 = 0          (2)  5^2x+1 - 26(5^x) + 5 = 0