Solving Exponential Equations
An exponential equation is an equation where one or more of the
exponents contains a variable. Some types of exponential equations can
be solved using the following property.
Property — Exponential Equality
If bx = by, then x = y. Here, b > 0 and b
≠ 1.
Note:
Recall that for a one-to-one
function, two different inputs
always result in two different
outputs.
That is, in a one-to-one function,
each output value corresponds to
exactly one input value.
This property is a direct consequence of the fact that exponential functions
are one-to-one functions. That is, if two exponential functions have the
same output, bx and by, then their inputs, x and y, must be equal.
Example 1
Solve: 25 5x-8 = 625
| Solution
First, write each expression using the same base, 5.
On the left, use the Power of a Power Property.
Use the Exponential Equality Property to set
the exponents equal to each other.
|
25 5x - 8 (52) 5x - 8
5 10x - 16
10x - 16 |
= 625 = 54
= 54
= 4 |
| Add 16 to both sides.
Divide both sides by 10. |
10x x |
= 20 = 2 |
We can check the solution by replacing x with 2 in the original equation
and simplifying.
| Original equation: |
Is
Is
Is |
25 5x - 8
25 5 · 2 - 8
252
625 |
= 625 = 625 ?
= 625 ?
= 625 ? Yes |
So, the solution of 25 5x - 8 = 625 is x = 2.
Note:
To write each base in exponential form,
first find its prime factorization.
For example:
25 = 5 · 5 = 52
625 = 5 · 5
· 5 · 5 = 54
The Power of a Power Property says
(xm)n 5 xmn.
Example 2
Solve: 163x + 1 = 324x
| Solution
Write each expression with the same base, 2.
Use the Power of a Power Property.
Use the Exponential Equality Property to set
the exponents equal to each other.
|
163x + 1 (24)3x + 1
212x + 4
12x + 4 |
= 324x = (25)4x
= 220x
= 20x |
| Subtract 12x from both sides. |
4 |
= 8x |
| Divide both sides by 8.
|
 |
= x |
We can check the solution in the usual way.
So, the solution of 163x + 1 = 324x is
Note:
This method can only be used if we
write each side of the equation as an
exponential expression using the same
base. If not, we must use a different
method.
For example, we can use this method
solve 2x = 8 but not to
solve 2x = 9.
Recall that
Thus,
|
