1. Equating the Powers when base is same

2. Introduction
If exponents have same base, then we can equate the powers ax = ab
Power (x)
Power (b)
Equating the base x = b
Same base (a)

3. Ans: x=3 Example 1: If 5x=53 , Find x? Solution
Bases(5) are equal so equating the powers, we get
Ans: x=3
Example 2: If 7x=75, find x?
7x = 75
Bases(7) are equal so equating the powers, we get
Ans: x=5 x 3 x 5

4. Bases(4) are equal so equating the powers, we get
Example 3: If 4x=64, find x
Solution – To find x
Step 1: To make base same
Step 2: Rewrite 64 as a power with base 4
Step 3: Factorize 64 with 4 till we get 1
4x = 64
4x = 43
Bases(4) are equal so equating the powers, we get
4x = 43
Ans: x=3
x = 3

5. 12x = 122 Example 4: If 12x=144, find x? Solution: To find x,
Step 1: To make base same
Step 2: Rewrite 144 as a power with base 12
Step 3: Factorize 144 with 12 till we get 1
12x = 144
12x = 122
Bases(12) are equal so equating the powers, we get
12x =
Ans: x=2
x =

6. Example 5: Solution: 2x + x = 6 3x = 6 x = Ans: x = 2 base is same If
, find x
Solution:
base is same
, equating the powers
2x+x =
2x + x = 6
3x = 6
x =
Ans: x = 2
= 2

7. Example 6: If
, find x
Solution:
To make it as a single base we apply the Quotient rule,
6x =

8. are same so equating the powers
Cont……..
6x - 3 = 21
6x = (Transposing -3 to +3)
6x = 24
x =
∴ x= 4
Bases
are same so equating the powers
(Transposing 6)

9. Example 7: Solution: If , find x
To make it as single power, we apply the power rule (am)n = amxn
6x =

10. Cont…… 6x=18 x = ∴ x = 3 Bases are equal so equating the powers
(Transposing 6)

11. Expressing a Number with a specific base
Example 1: Express 32 as a power with the base 2
Solution:
Factorize 32 with 2 till we get 1
32 = 2 x 2 x 2 x 2 x 2 = 25
32 = 25
2x = 25
Bases(2) are equal so equating the powers, we get
Ans: x = 5
x = 5

12. Example 2: Express 162 as a power with the base 4 Solution:
Factorize 16 with 4 till we get 1
162 = (4 x 4)2 = (42)2
162 = (42)2
To make it as single power, we apply the power rule (am)n = amxn
(42)2 = 42x2 = 44
4x = 44
Bases(4) are equal so equating the powers, we get
Ans: x = 4
x = 4

13. (52)3 = 52 x 3 = 56 Example 3: Express 253 as a power with the base 5
Solution:
Factorize 25 with 5 till we get 1
253 = (5 x5)3 = (52)3
253 = (52)3
To make it as single power, we apply the power rule (am)n = amxn
(52)3 = 52 x 3 = 56
5x = 56
Bases(5) are equal so equating the powers, we get
Ans: x = 6
x = 6

14. Try these If 7x=75 , find x If 11x=121, find x
Express 64 as a power with base 4
Express 492 as a power with base 7