Solve: [tex]6^{2 x-3}=6^{-2 x+1}[/tex]
Answer (1)
Equate the exponents: 2 x − 3 = − 2 x + 1 .
Add 2 x to both sides: 4 x − 3 = 1 .
Add 3 to both sides: 4 x = 4 .
Divide by 4 to find the solution: x = 1 .
Explanation
Understanding the Problem We are given the equation 6 2 x − 3 = 6 − 2 x + 1 . Our goal is to solve for x . Since the bases are the same, we can equate the exponents.
Equating Exponents Equating the exponents, we have: 2 x − 3 = − 2 x + 1
Adding 2x to Both Sides Now, we want to isolate x . First, add 2 x to both sides of the equation: 2 x + 2 x − 3 = − 2 x + 2 x + 1 4 x − 3 = 1
Adding 3 to Both Sides Next, add 3 to both sides of the equation: 4 x − 3 + 3 = 1 + 3 4 x = 4
Dividing by 4 Finally, divide both sides by 4: 4 4 x = 4 4 x = 1
Final Answer Therefore, the solution to the equation 6 2 x − 3 = 6 − 2 x + 1 is x = 1 .
Examples
Imagine you're adjusting the settings on a machine to ensure two processes run at the same rate. The equation 6 2 x − 3 = 6 − 2 x + 1 is similar to balancing these rates. By solving for x , you find the exact setting needed to equalize the processes, ensuring smooth operation. This type of exponential equation can be applied in various fields, such as engineering, finance, and computer science, to model and solve problems involving growth, decay, and equilibrium.
