Solve: [tex]$3,125=5^{-10+3 x}$[/tex]
Answer (2)
To solve the equation 3125 = 5 − 10 + 3 x , we rewrite 3125 as 5 5 , leading to the equation 5 5 = 5 − 10 + 3 x . By equating the exponents, we find that 5 = − 10 + 3 x , which simplifies to give us the solution x = 5 . Thus, the final answer is x = 5 .
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Rewrite 3125 as 5 5 , so the equation becomes 5 5 = 5 − 10 + 3 x .
Equate the exponents: 5 = − 10 + 3 x .
Solve for x : 3 x = 15 .
The solution is x = 5 .
Explanation
Problem Analysis We are given the equation 3125 = 5 − 10 + 3 x and we need to solve for x .
Expressing both sides with the same base First, we express 3125 as a power of 5. Since 3125 = 5 5 , we can rewrite the equation as: 5 5 = 5 − 10 + 3 x
Equating the exponents Since the bases are equal, we can equate the exponents: 5 = − 10 + 3 x
Isolating the term with x Now, we solve for x . Add 10 to both sides of the equation: 5 + 10 = − 10 + 3 x + 10 15 = 3 x
Solving for x Divide both sides by 3: 3 15 = 3 3 x 5 = x So, x = 5 .
Final Answer Therefore, the solution to the equation 3125 = 5 − 10 + 3 x is x = 5 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to model compound interest. If you invest a principal amount P at an annual interest rate r compounded n times per year, the amount A you will have after t years is given by A = P ( 1 + n r ) n t . Solving for t in such equations involves logarithms, which are closely related to exponential functions. Understanding how to solve exponential equations is crucial for making informed financial decisions.
