1. Solving Exponential Equations
6.5
Solving Exponential Equations
How can you solve an exponential equation algebraically?
Students will be able to solve exponential equations with the same base.
Students will be able to solve exponential equations with unlike bases.

2. Students will be able to solve exponential equations with the same base.
Solving Exponential Equations with the Same Base
Property of Equality for Exponential Equations
Two powers with the same positive base b, where 𝑏≠1, are equal if and only if their exponents are equal.
If 2 𝑥 = 2 5 , then 𝑥=5.
If 𝑏>0 and 𝑏≠1, then 𝑏 𝑥 = 𝑏 𝑦 if and only if 𝑥=𝑦.

3. Students will be able to solve exponential equations with the same base.
Solving Exponential Equations with the Same Base
Solve each equation
3 𝑥+1 = 3 5
𝑥+1=5
𝑥=4
6= 6 2𝑥−3
1=2𝑥−3
4=2𝑥
2=𝑥
10 3𝑥 = 10 2𝑥+3
3𝑥=2𝑥+3
𝑥=3
Since the bases are the same the exponents must equal each other.
Since the bases are the same the exponents must equal each other.
Since the bases are the same the exponents must equal each other.

4. You Try!!
Solve each equation
2 2𝑥 = 2 6
2𝑥=6
𝑥=3
5 2𝑥 = 5 𝑥+1
2𝑥=𝑥+1
𝑥=1
7 3𝑥+5 = 7 𝑥+1
3𝑥+5=𝑥+1
2𝑥+5=1
2𝑥=−4
𝑥=−2

5. 2. Students will be able to solve exponential equations with unlike bases.
Solving Exponential Equations with Unlike Bases
To solve some exponential equations, you must first rewrite each side of the equation using the same base.
5 𝑥 =125
5 𝑥 = 5 3
𝑥=3
4 𝑥 = 2 𝑥−3
( 2 2 ) 𝑥 = 2 𝑥−3
2 2𝑥 = 2 𝑥−3
2𝑥=𝑥−3
𝑥=−3
9 𝑥+2 = 27 𝑥
( 3 2 ) 𝑥+2 = ( 3 3 ) 𝑥
3 2(𝑥+2) = 3 3𝑥
3 2𝑥+4 = 3 3𝑥
2𝑥+4=3𝑥
4=𝑥
Since the bases are not the same can you write 125 as some power of 5?
Since the bases are not the same can you write 4 as some power of 2?
Since the bases are not the same can you write 9 and 27 as the power of the same number?
Powers of 3!

6. You Try!!
Solve each equation
4 𝑥 = 32 6
( 2 2 ) 𝑥 = ( 2 5 ) 6
2 2𝑥 = 2 30
2𝑥=30
𝑥=15
9 2𝑥 = 3 𝑥−6
( 3 2 ) 2𝑥 = 3 𝑥−6
3 4𝑥 = 3 𝑥−6
4𝑥=𝑥−6
3𝑥=−6
𝑥=−2
4 3𝑥 = 8 𝑥+1
( 2 2 ) 3𝑥 = ( 2 3 ) 𝑥+1
2 6𝑥 = 2 3𝑥+3
6𝑥=3𝑥+3
3𝑥=3
𝑥=1

7. 2. Students will be able to solve exponential equations with unlike bases.
More Solving Exponential Equations with Unlike Bases
To solve some exponential equations, you must first rewrite each side of the equation using the same base.
( ) 𝑥 =4
( 2 −1 ) 𝑥 = 2 2
2 −𝑥 = 2 2
−𝑥=2
𝑥=−2
4 𝑥+1 = 1 64
4 𝑥+1 = 4 −3
𝑥+1=−3
𝑥=−4
Since the bases are not the same can you write and 4 as the power of the same number?
Powers of 2!
Since the bases are not the same can you write 4 and as the power of the same number?
Powers of 4!

8. You Try!!
Solve the equation
( ) 𝑥−1 =27
( 3 −1 ) 𝑥−1 = 3 3
3 −𝑥+1 = 3 3
−𝑥+1=3
−𝑥=2
𝑥=−2

9. Let’s Review
Students will be able to solve exponential equations with the same base.
This is the easy one. If the bases are already the same the exponents must equal.
Then solve for x.
Students will be able to solve exponential equations with unlike bases.
This is the harder one. If the bases are not the same you have to think, what number raised to a power will equal the original base.
You want to have both bases changed to the same base.
Then the exponents must equal.