Math  /  Algebra

QuestionSolve the exponential equation by expressing each side as a power of the same base and then equating the exponents. 94x12=65619^{4 x-12}=6561
The solution set is \square B.

Studdy Solution

STEP 1

1. The equation 94x12=6561 9^{4x - 12} = 6561 is an exponential equation.
2. We need to express both sides of the equation as powers of the same base to solve for x x .

STEP 2

1. Express both sides of the equation as powers of the same base.
2. Equate the exponents.
3. Solve for x x .

STEP 3

First, express both sides of the equation as powers of the same base. Notice that 9 9 can be written as 32 3^2 and 6561 6561 can be written as 38 3^8 :
94x12=6561 9^{4x - 12} = 6561 (32)4x12=38 (3^2)^{4x - 12} = 3^8

STEP 4

Use the property of exponents (am)n=amn(a^m)^n = a^{m \cdot n} to simplify the left side of the equation:
32(4x12)=38 3^{2(4x - 12)} = 3^8 38x24=38 3^{8x - 24} = 3^8

STEP 5

Since the bases are the same, equate the exponents:
8x24=8 8x - 24 = 8

STEP 6

Solve for x x by first adding 24 to both sides:
8x24=8 8x - 24 = 8 8x=8+24 8x = 8 + 24 8x=32 8x = 32
Now, divide both sides by 8:
x=328 x = \frac{32}{8} x=4 x = 4
The solution set is:
4 \boxed{4}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord