Exponential equation is an equation whose exponent or part of the exponent is a variable.
Examples :
2x = 8
73x - 1 = 49
The following steps would be useful to solve exponential equations by rewriting the base.
Step 1 :
Using the rules of exponents, rewrite each side of the equation as a power with the same base.
Step 2 :
Once you get the same base on both sides in step 1, equate the exponents and solve for the variable.
ax = ak
x = k
Solve for x in each of the following :
Example 1 :
2x = 32
Solution :
2x = 32
2x = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2
2x = 25
Equate the exponents.
x = 5
Example 2 :
3x - 2 = 1/9
Solution :
3x - 2 = 1/9
3x - 2 = 1/32
3x - 2 = 3-2
Equate the exponents.
x - 2 = -2
Add 2 to both sides.
x = 0
Example 3 :
6 ⋅ 3x = 54
Solution :
6 ⋅ 3x = 54
Divide both sides by 6.
3x = 54/6
3x = 9
3x = 32
Equate the exponents.
x = 2
Example 4 :
4x - 1 = (1/2)1 - 3x
Solution :
4x - 1 = (1/2)1 - 3x
22(x - 1) = 2-1(1 - 3x)
22x - 2 = 2-1 + 3x
Equate the exponents.
2x - 2 = -1 + 3x
Subtract 3x from both sides.
-x - 2 = -1
Add 2 to both sides.
-x = 1
Multiply both sides by -1.
x = -1
Example 5 :
2x + 2 = 1/4
Solution :
2x + 2 = 1/4
2x + 2 = 1/22
2x + 2 = 2-2
Equate the exponents.
x + 2 = -2
Subtract 2 from both sides.
x = -4
Example 6 :
31 - 2x = 1/27
Solution :
31 - 2x = 1/27
31 - 2x = 1/33
31 - 2x = 3-3
Equate the exponents.
1 - 2x = -3
Subtract 1 from both sides.
-2x = -4
Divide both sides by -2.
x = 2
Example 7 :
5 ⋅ 2x = 40
Solution :
5 ⋅ 2x = 40
Divide both sides by 5.
2x = 40/5
2x = 8
2x = 23
Equate the exponents.
x = 3
Example 8 :
9x = 7(3x) + 18
Solution :
9x = 7(3x) + 18
(32)x = 7(3x) + 18
(3x)2 = 7(3x) + 18
(3x)2 - 7(3x) - 18 = 0
Let a = 3x.
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
Example 9 :
3x + 1 ⋅ 9-x = (1/3)x + 1
Solution :
3x + 1 ⋅ 9-x = (1/3)x + 1
3x + 1 ⋅ (32)-x = (3-1)x + 1
3x + 1 ⋅ 3-2x = 3-1(x + 1)
3x + 1 - 2x = 3-x - 1
3-x + 1 = 3-x - 1
Equate the exponents.
-x + 1 = -x - 1
Add x to both sides.
1 = -1
In the final step solving the given equation, the variable is no more. And also, 1 = -1 is a false statement, so, there is no solution for this equation.
Example 10 :
2x ⋅ 42 - x = 8
Solution :
2x ⋅ 42 - x = 8
2⋅ (22)2 - x = 23
21⋅ 24 - 2x = 23
21 + 4 - 2x = 23
25 - 2x = 23
Equate the exponents.
5 - 2x = 3
Subtract 5 from both sides.
-2x = -2
Divide both sides by -2.
x = 1
Example 11 :
2x - 1⋅ 42x + 1 = 32
Solution :
2x -1⋅ 42x + 1 = 32
2x - 1⋅ (22)2x + 1 = 25
2x - 1⋅ 22(2x + 1) = 25
2x - 1⋅ 24x + 2 = 25
2x - 1 + 4x + 2 = 25
25x + 1 = 25
Equate the exponents.
5x + 1 = 5
Subtract 1 from both sides.
5x = 4
Divide both sides by 5.
x = 4/5
Example 12 :
53x - 2 = 1252x
Solution :
53x - 2 = 1252x
53x - 2 = (53)2x
53x - 2 = 56x
Equate the exponents.
3x - 2 = 6x
Subtract 6x from both sides.
-3x - 2 = 0
Add 2 to both sides.
-3x = 2
Divide both sides by -3.
x = -2/3
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