Exponential Equation
How to solve an exponential equation: 5 examples and their solutions.
Example 1
Example
Solution
To solve an exponential equation,
make the bases of both sides the same.
The base of the left side 23x - 1 is 2.
So change the base of the right side to 2:
4 = 22.
The bases of both sides are the same.
Then the exponents of both sides are the same.
So 3x - 1 = 2.
Solve 3x - 1 = 2.
Move -1 to the right side.
Divide both sides by 3.
Then x = 1.
So x = 1.
Example 2
Example
Solution
The base of the left side 3x - 5 is 3.
So change the base of the right side to 3.
9 = 32
(32)4 = 32⋅4 = 38
Power of a Power
3x - 5 = 38
The bases of both sides are the same.
Then the exponents of both sides are the same.
So x - 5 = 8.
Solve x - 5 = 8.
Move -5 to the right side.
Then x = 13.
So x = 13.
Example 3
Example
Solution
The base of the left side 74x + 8 is 7.
So change the base of the right side to 7.
1 = 70
Zero Exponent
The bases of both sides are the same.
Then the exponents of both sides are the same.
So 4x + 8 = 0.
Solve 4x + 8 = 0.
Move +8 to the right side.
Divide both sides by 4.
Then x = -2.
So x = -2.
Example 4
Example
Solution
The bases of the numbers are 125, 5, and 1/25.
So change the bases of the numbers to 5.
125 = 53
(1/25)x = 25-x
Negative Exponent
53⋅53 = 53 + x
Product of Powers
25-x = (52)-x
(52)-x = 5-2x
53 + x = 5-2x
The bases of both sides are the same.
Then the exponents of both sides are the same.
So 3 + x = -2x.
Solve 3 + x = -2x.
Move 3 to the right side.
Move -2x to the left side.
Then, x + 2x, 3x is equal to -3.
Divide both sides by 3.
Then x = -1.
So x = -1.
Example 5
Example
Solution
The given equation has 3 terms,
not 2 terms.
So you can't directly compare the exponents
like the previous examples.
To solve this equation,
first move the right side terms to the left side.
4x = (22)x = 22x
22x = (2x)2
Think 2x as a variable
and factor (2x)2 x - 3⋅2x - 4 = 0.
Find a pair of numbers
whose product is the constant term -4
and whose sum is the coefficient of the middle term -3.
-4⋅1 = -4
-4 + 1 = -3
So (2x - 4)(2x + 1) = 0.
Factor a Quadratic Trinomial
Move -4 to the right side.
Then 2x = 4.
Change the base of 4 to 2.
4 = 22
2x = 22
So x = 2.
This is the answer for case 1.
Case 2: 2x + 1 = 0
Move +1 to the right side.
Then 2x = -1.
See 2x = -1.
The left side 2x is always plus.
(The graph of y = ax is always above the x-axis.)
But the right side -1 is minus.
So this equation has no solution.
This is the answer for case 2.
For case 1, x = 2.
For case 2, there's no solution.
So the solution of (2x - 4)(2x + 1) = 0 is
x = 2.
So x = 2 is the answer.