Solve: [tex]49^{3 x}=343^{2 x+1}[/tex]
Answer (1)
Rewrite both sides of the equation with the same base: 4 9 3 x = ( 7 2 ) 3 x and 34 3 2 x + 1 = ( 7 3 ) 2 x + 1 .
Simplify the exponents: 7 6 x = 7 6 x + 3 .
Equate the exponents: 6 x = 6 x + 3 .
Solve for x: Since 0 = 3 is a contradiction, there is no solution. no solution
Explanation
Problem Analysis We are given the equation 4 9 3 x = 34 3 2 x + 1 and asked to solve for x . We will rewrite both sides of the equation with the same base.
Rewriting with Common Base Since 49 = 7 2 and 343 = 7 3 , we can rewrite the equation as ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplifying Exponents Using the power of a power rule, we simplify the exponents: 7 6 x = 7 6 x + 3 .
Equating Exponents Since the bases are equal, the exponents must be equal: 6 x = 6 x + 3 .
Solving for x Subtracting 6 x from both sides, we get 6 x − 6 x = 3 , which simplifies to 0 = 3 .
Conclusion Since 0 = 3 is a contradiction, there is no solution to the equation.
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 1000 inana cco u n tt ha tp a ys 5 A = 1000(1.05)^t$. Solving exponential equations allows you to determine how long it will take for your investment to reach a certain value.
