Indicial Exponential Equation

CONTENT

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

Exponential Equation of Linear Form

Under 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 equation 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