Solve: [tex]49^{3 x}=343^{2 x+1}[/tex]
Answer (2)
To solve the equation 4 9 3 x = 34 3 2 x + 1 , we express both sides with a common base. This results in an equation that leads to a contradiction, showing that there is no solution. Thus, the equation has no solutions.
;
Express both sides of the equation with the same base: 49 = 7 2 and 343 = 7 3 , so the equation becomes ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the equation using the power of a power rule: 7 6 x = 7 6 x + 3 .
Equate the exponents: 6 x = 6 x + 3 .
Solve for x : 0 = 3 , which is a contradiction. Therefore, there is no solution .
Explanation
Analyze the problem We are given the equation 4 9 3 x = 34 3 2 x + 1 and need to find the value of x that satisfies this equation. We will express both sides of the equation using the same base to simplify it.
Express with the same base First, we express 49 and 343 as powers of 7. We know that 49 = 7 2 and 343 = 7 3 . Substituting these into the original equation, we get ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the equation Using the power of a power rule, which states that ( a m ) n = a mn , we simplify both sides of the equation: 7 2 ( 3 x ) = 7 3 ( 2 x + 1 ) , which simplifies to 7 6 x = 7 6 x + 3 .
Equate the exponents Since the bases are equal, we can equate the exponents: 6 x = 6 x + 3 .
Solve for x Now, we solve for x . Subtracting 6 x from both sides, we get 0 = 3 . This is a contradiction, meaning there is no value of x that satisfies the original equation.
Final Answer Therefore, the equation 4 9 3 x = 34 3 2 x + 1 has no solution.
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 an amount of money P at an annual interest rate r , compounded annually. After t years, the amount of money you have is given by A = P ( 1 + r ) t . Solving exponential equations allows you to determine how long it will take for your investment to reach a certain value.
