Use the equation $2,401=7^{6-2 x}$. What power of 7 is 2,401?
Answer (2)
Recognize that the question asks for the exponent y such that 7 y = 2401 .
Determine that 7 4 = 2401 .
Conclude that the power of 7 that equals 2401 is 4.
The final answer is 4 .
Explanation
Understanding the Problem We are given the equation 2401 = 7 6 − 2 x and asked to find the power of 7 that equals 2401. In other words, we want to find the value of y such that 7 y = 2401 .
Finding the Power of 7 First, let's determine what power of 7 equals 2401. We can test the given options or use logarithms. We find that 7 4 = 2401 .
Equating Exponents Now we know that 2401 = 7 4 . We can substitute this into the given equation: 7 4 = 7 6 − 2 x . Since the bases are equal, the exponents must be equal. Therefore, 4 = 6 − 2 x .
Solving for x Next, we solve for x . Subtract 6 from both sides: 4 − 6 = − 2 x , which simplifies to − 2 = − 2 x . Divide both sides by -2: x = 1 .
Finding the Power of 7 that Equals 2401 We are asked to find the power of 7 that equals 2401. From our earlier calculation, we found that 7 4 = 2401 . Therefore, the power of 7 that equals 2401 is 4.
Examples
Understanding exponents is crucial in many fields, such as finance when calculating compound interest. For example, if you invest 1000 a t anann u a l in t eres t r a t eo f 5 t ye a rs i s g i v e nb y A = 1000(1.05)^t . De t er minin g h o wl o n g i tt ak es t o d o u b l eyo u r in v es t m e n t in v o l v esso l v in g f or t$ in the equation 2000 = 1000 ( 1.05 ) t , which uses similar exponential principles.
The power of 7 that equals 2401 is 4, as we determined that 7 4 = 2401 . We derived this from the equation 2401 = 7 6 − 2 x and solved for the exponent. Therefore, the final answer is 4 .
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