Explanation:
The first technique we will introduce for solving exponential equations involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers b, S, and T, where
b
>
0
,
b
≠
1
,
b
S
=
b
T
if and only if S = T.
In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
For example, consider the equation
3
4
x
−
7
=
3
2
x
3
. To solve for x, we use the division property of exponents to rewrite the right side so that both sides have the common base 3. Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for x:
3
4
x
−
7
=
3
2
x
3
3
4
x
−
7
=
3
2
x
3
1
Rewrite 3 as 3
1
.
3
4
x
−
7
=
3
2
x
−
1
Use the division property of exponents
.
4
x
−
7
=
2
x
−
1
Apply the one-to-one property of exponents
.
2
x
=
6
Subtract 2
x
and add 7 to both sides
.
x
=
3
Divide by 3
.
A GENERAL NOTE: USING THE ONE-TO-ONE PROPERTY OF EXPONENTIAL FUNCTIONS TO SOLVE EXPONENTIAL EQUATIONS
For any algebraic expressions S and T, and any positive real number
b
≠
1
,
b
S
=
b
T
if and only if
S
=
T
HOW TO: GIVEN AN EXPONENTIAL EQUATION OF THE FORM
b
S
=
b
T
, WHERE S AND T ARE ALGEBRAIC EXPRESSIONS WITH AN UNKNOWN, SOLVE FOR THE UNKNOWN
Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form
b
S
=
b
T
.
Use the one-to-one property to set the exponents equal to each other.
Solve the resulting equation, S = T, for the unknown.