Solve $2,401=7^{6-2 x}$
Answer (1)
Rewrite 2401 as 7 4 .
Equate the exponents: 4 = 6 − 2 x .
Solve for x : − 2 = − 2 x .
The solution is 1 .
Explanation
Problem Analysis We are given the equation 2401 = 7 6 − 2 x and we need to solve for x .
Expressing 2401 as a power of 7 First, we express 2401 as a power of 7. Since 7 4 = 2401 , we can rewrite the equation as 7 4 = 7 6 − 2 x .
Equating the exponents Now, because the bases are equal, we can equate the exponents: 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 .
Finding the value of x Finally, divide both sides by -2: x = − 2 − 2 = 1 . Therefore, x = 1 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest calculations involve exponential growth. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Solving for t or r in such equations involves similar logarithmic and algebraic techniques as used in the given problem.
