INDICIAL AND EXPONENTIAL EQUATIONS

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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

1. 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

1. 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

1. 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

1. 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

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)2^2x – 6 (2^x) + 8 = 0
2. b)5^2x + 4 x 5 ^x+1 – 125 = 0
3. c)3^2x – 9 = 0

Solution

1. 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

 

1. 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

Evaluation

Solvethe following exponential equations

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

 

Exponential Equation of Quadratic Form

Someexponential equation can be reduced to quadratic form as can be seen below.

Example:

Solvethe following equations.

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

 

Solution

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

 

1. 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

 

1. 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

Solvethe following exponential equations.

1. a)         2^{2x+ 1}– 5 (2^x)+ 2 = 0
2. b)         3^2x– 4 (3^x+1) +27 = 0

 

Reading Assignment: Further Mathematics Project Book 1(New thirdedition).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

Solvethe following exponential equations

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

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