Solve: [tex]3,125=5^{-10+3 x}[/tex]
[tex]x=\square[/tex]
Answer (1)
Express 3125 as 5 5 .
Equate the exponents: 5 = − 10 + 3 x .
Solve the linear equation for x : 15 = 3 x .
Find the value of x : 5 .
Explanation
Understanding the Problem We are given the equation 3125 = 5 − 10 + 3 x and we need to solve for x .
Expressing 3125 as a power of 5 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 Now, since the bases are equal, we can equate the exponents: 5 = − 10 + 3 x .
Solving for x Next, we solve the linear equation 5 = − 10 + 3 x for x . Add 10 to both sides of the equation: 5 + 10 = − 10 + 3 x + 10 , which simplifies to 15 = 3 x .
Finding the value of x Finally, divide both sides by 3: 3 15 = 3 3 x , which gives us 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. Suppose 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 exponential equations helps determine how long it will take for your investment to reach a certain value.
