Equating the Powers when base is same

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)

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

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

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 = 122 Ans: x=2 x = 2

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 = 6 2x + x = 6 3x = 6 x = Ans: x = 2 = 2

Example 6: If , find x Solution: To make it as a single base we apply the Quotient rule, 6x-3 = 21

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

Example 7: Solution: If , find x To make it as single power, we apply the power rule (am)n = amxn 6x = 18

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

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

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

(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

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