Exponential Equations

Exponential Equations

To solve exponential equations without logarithms, you need to have equations with comparable exponential expressions on either side of the "equals" sign.  Then you can compare the powers and solve.

Note that if ar = as, then r = s. If there is a way to rewrite expressions with like bases, the exponents of those bases will then be equal to one another.

Like Bases

When the bases are the same, the powers must also be the same for the equation to be true.
You can set the powers equal to each other and solve the resulting equation to find an unknown value.

Example

10x = 103
x = 3

Example

101-x = 106

1 - x = 6
1 - 6 = x
-5 = x

Converting a Base

When the bases are not the same, you will need to convert one or both bases so that they are the same.  This often requires some familiarity with squares, cubes, and higher powers of numbers one through nine.

Example

56x+1 = 625

56x+1 = 54

6x + 1 = 4
6x = 3
x = 1/2

Converting Both Bases

Sometimes, both bases may need to be converted in order to match.

Example

42x+2 = 8

Rewrite the problem using the common base of 2:

4 = 22
8 = 23

(22)(2x+2) = 23

Simplify powers raised to a power by multiplying the exponents.

2(2)(2x+2) = 23

24x + 4 = 23

Once the bases are equal, the exponents may be set equal to one another and the equation solved.

4x + 4 = 3
4x = -1
x = -1/4

Working with Fractions and Negative Exponents

Negative exponents indicate that a base belongs in the denominator of a fraction.

For instance, 1/25 = 5-2

Example

53x+1 = 1/25

53x+1 = 5-2

3x + 1 = -2
3x = -3
x = -1

Working with Radical Signs (Square Roots)

Note that a square root of a number is the same as the base raised to power of one-half.

Example

62x-2 = 61/2

2x - 2 = 1/2
2x = 5/2
x = 5/4

Working with Exponents Raised to a Power

Remember that when a power is raised to a power, the exponents are multiplied.

(xa)b = xab

A power applied to a product within parentheses affects each element inside the parentheses.

(xy3)2 = x2y6

However, be careful to note that exponents do not "distribute" across addition.

(x + y2)3 does not equal x3 + y6

Example

x2 - 3x = 4
x2 - 3x - 4 = 0
(x -4)(x +1) = 0
x = {-1, 4}