To solve exponential equations, we have to follow the steps given below.
(i) Arrange the terms which are having exponents as x.
(ii) Change bases on both sides as equal.
(iii) If bases are equal on both sides, we can equate powers.
(iv) If powers are equal, then equate the bases.
Example 1 :
Solve 4x+1 = 32
Solution :
4x+1 = 32
On both sides, bases are multiples of 2.
(22)x+1 = 32
2 2(x+1) = 25
Since on both sides the bases are equal, we can equate the powers.
2(x+1) = 5
x+1 = 5/2
x = (5/2)-1
x = 3/2
Example 2 :
Solve (1/27)4-x = 92x-1
Solution :
(1/27)4-x = 92x-1
On both sides of the equal sign, bases are multiples of 3.
(1/33)4-x = (32)2x-1
(3-3)4-x = (32)2x-1
3-12+3x = 34x-2
-12+3x = 4x-2
3x-4x = -2+12
-x = 10
x = -10
Example 3 :
Solve (813x+2)/243x = 3
Solution :
(813x+2)/243x = 3
(34)3x+2/(35)x = 3
312x+8/35x = 3
312x+8-5x = 31
37x+8 = 31
Equating bases, we get
7x+8 = 1
7x = -7
x = -1
Example 4 :
Solve (1/6)3x+6⋅ 2163x = 1/216
Solution :
(1/6)3x+6⋅ 2163x = 1/216
(6-1)3x+6⋅ (63)3x = (1/6)3
6-3x-6⋅ 69x = 6-3
6-3x-6+9x = 6-3
6-6+6x = 6-3
6x-6 = -3
6x = -3+6
6x = 3
x = 3/6
x = 1/2
Example 5 :
Solve 2(42x+1) = 128
Solution :
2(42x+1) = 128
2(22(2x+1)) = 128
22(2x+1) = 64
24x+2 = 26
4x+2 = 6
4x = 4
x = 1
Example 6 :
Solve √5 = 5x+4
Solution :
√5 = 5x+4
51/2 = 5x+4
Equating powers, we get
1/2 = x+4
x = (1/2) - 4
x = -7/2
Example 7 :
Solve 2x+21-x = 3
Solution :`
2x+21-x = 3
2x+(21/2x) = 3
((2x)2+2)/2x = 3
a2+2 = 3a
a2-3a+2 = 0
(a-1)(a-3) = 0
a = 1 and a = 3
Example 8 :
Solve 72x+1 = 1/49
Solution :
72x+1 = 1/49
72x+1 = 7-2
2x + 1 = -2
2x = -2-1
2x = -3
x = -3/2
Example 9 :
Solve 32x+3-1/3 = 8/3
Solution :
32x+3-1/3 = 8/3
32x+3 = 8/3 + 1/3
32x+3 = 3
2x+3 = 1
2x = -2
x = -1
Example 10 :
Solve 10-3x ⋅ 10x = 1/10
Solution :
10-3x ⋅ 10x = 1/10
10-3x+x = 10-1
-2x = -1
x = 1/2
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