To solve $49^{3 x}=343^{2 x+1}$, write each side of the equation in terms of base $\square$.
Answer (1)
Express 49 as 7 2 and 343 as 7 3 .
Rewrite the equation as ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the exponents to get 7 6 x = 7 6 x + 3 .
Write each side of the equation in terms of base 7: 7 .
Explanation
Analyze the problem We are given the equation 4 9 3 x = 34 3 2 x + 1 . Our goal is to rewrite both sides of the equation using the same base. We recognize that both 49 and 343 are powers of 7.
Rewrite with base 7 We can express 49 as 7 2 and 343 as 7 3 . Substituting these into the original equation, we get ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the exponents Using the power of a power rule, which states that ( a m ) n = a mn , we simplify the exponents: 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 and answer the question Subtracting 6 x from both sides, we get 0 = 3 , which is a contradiction. This means there is no solution for x that satisfies the original equation. However, the question asks us to write each side of the equation in terms of base 7. We have already done this in step 3: 4 9 3 x = ( 7 2 ) 3 x = 7 6 x and 34 3 2 x + 1 = ( 7 3 ) 2 x + 1 = 7 6 x + 3 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest calculations involve exponential growth. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A you will have after t years is given by A = P ( 1 + / n ) n t . Understanding how to manipulate exponential equations allows you to calculate the future value of your investments or the time it takes for an investment to reach a certain value.
