Solve: [tex]4^{3 x}=4^2[/tex]
Answer (2)
The solution to the equation 4 3 x = 4 2 is x = 3 2 after equating the exponents from both sides since they have the same base. The process involves isolating x by dividing both sides by 3. Therefore, the final answer is 3 2 .
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The problem provides an exponential equation with the same base on both sides.
Equate the exponents: 3 x = 2 .
Solve for x by dividing both sides by 3: x = 3 2 .
The solution to the equation is 3 2 .
Explanation
Understanding the Problem We are given the equation 4 3 x = 4 2 . Our goal is to find the value of x that satisfies this equation. Since the bases are the same (both are 4), we can equate the exponents.
Equating the Exponents Now, we set the exponents equal to each other: 3 x = 2
Solving for x To solve for x , we divide both sides of the equation by 3: 3 3 x = 3 2 This simplifies to: x = 3 2
Final Answer Therefore, the solution to the equation 4 3 x = 4 2 is x = 3 2 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For instance, they can model population growth, radioactive decay, and compound interest. Imagine you invest money in a bank account that offers continuous compounding interest. The amount of money you'll have after a certain time can be calculated using an exponential equation, helping you predict the growth of your investment.
