Solve for x in each of the following exponential equations.
Problem 1 :
2x = 4
Problem 2 :
4x = 1/2
Problem 3 :
9x = 1/27
Problem 4 :
36x = 1/6
Problem 5 :
5x = 1
Problem 6 :
8x + 1 = 16
Problem 7 :
82 - x = 1/64
Problem 8 :
(1/2)x - 1 = 4
Problem 9 :
(1/3)x - 2 = 1/9
Problem 10 :
2x - 1 = 5
Problem 11 :
5x - 1 - 2x = 0
Problem 12 :
5x = (√25)-7 ⋅ (√5)-5)
Problem 13 :
4x = 7(2x) + 18
1. Answer :
2x = 4
2x = 22
x = 2
2. Answer :
4x = 1/2
(22)x = 1/21
22x = 2-1
2x = -1
Divide both sides by 2.
x = -1/2
3. Answer :
9x = 1/27
(32)x = 1/33
32x = 3-3
2x = -3
Divide both sides by 2.
x = -3/2
4. Answer :
36x = 1/6
(62)x = 1/61
62x = 6-1
2x = -1
Divide both sides by 2.
x = -1/2
5. Answer :
5x = 1
5x = 50
x = 0
6. Answer :
8x + 1 = 16
(23)x + 1 = 24
23x + 1 = 24
3x + 1 = 4
Subtract 1 from both sides.
3x = 3
Divide both sides by 3.
x = 1
7. Answer :
82 - x = 1/64
82 - x = 1/82
82 - x = 8-2
2 - x = -2
Subtract 2 from both sides.
-x = -4
Multiply both sides by -1.
x = 4
8. Answer :
(1/2)x - 1 = 4
(2-1)x - 1 = 22
2-1(x - 1) = 22
2-x + 1 = 22
-x + 1 = 2
Subtract 1 from both sides.
-x = 1
Multiply both sides by -1.
x = -1
9. Answer :
(1/3)x - 2 = 1/9
(3-1)x - 2 = 32
3-1(x - 2) = 32
3-x + 2 = 32
-x + 2 = 9
Subtract 2 from both sides.
-x = 7
Multiply both sides by -1.
x = -7
10. Answer :
2x - 1 = 5
Add 1 to both sides.
2x = 5
In the given equation, we have the base 2 on the left side and 5 on the right side is not a power of 2. Since we don't have the same base on both sides, we can take logarithm with any base on both sides and solve for x.
log(2x) = log6
Using power rule of logarithm,
xlog2 = log6
Divide both sides by log2.
11. Answer :
5x - 1 - 2x = 0
Add 2x to both sides.
5x - 1 = 2x
In the equation above, 5 is not a power of 2 and also 2 is not a power of 5. Since we don't have the same base on both sides, we can take logarithm with any base on both sides and solve for x.
log(5x - 1) = log(2x)
Using power rule of logarithm,
(x - 1)log5 = xlog2
Using distributive property,
xlog5 - log5 = xlog2
Subtract xlog2 from both sides.
xlog5 - log5 - xlog2 = 0
Add log5 to both sides.
xlog5 - xlog2 = log5
Factor.
x(log5 - log)2 = log5
Divide both sides by (log5 - log2).
12. Answer :
5x = (√25)-7 ⋅ (√5)-5
5x = 5-7 ⋅ (51/2)-5
5x = 5-7 ⋅ 5-5/2
5x = 5-7 - 5/2
5x = 5-19/2
Equate the exponents.
x = -19/2
13. Answer :
4x = 7(2x) + 18
(22)x = 7(2x) + 18
(2x)2 = 7(2x) + 18
(2x)2 - 7(2x) - 18 = 0
Let a = 2x.
a2 - 7a - 18 = 0
(a - 9)(a + 2) = 0
a - 9 = 0 or a + 2 = 0
a - 9 = 0 a = 9 a = 32 3x = 32 x = 2 |
a + 2 = 0 a = -2 a = -2 3x = -2 |
In 3x, whatever real value (positive or negative or zero) we substitute for x, 3x can never be negative. So we can ignore the equation 3x = -2.
Therefore,
x = 2
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