To solve $49^{3 x}=343^{2 x+1}$, write each side of the equation in terms of base $\square$.

The equation 4 9 3 x = 34 3 2 x + 1 can be rewritten in terms of base 7 as 7 6 x = 7 6 x + 3 . This leads to a contradiction, indicating there is no solution for x . The problem illustrates the importance of using a common base in exponential equations.
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Answered by Anonymous | 2025-07-04

Express 49 and 343 as powers of 7: 49 = 7 2 and 343 = 7 3 .
Rewrite the equation using base 7: ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the exponents: 7 6 x = 7 6 x + 3 .
The equation in terms of base 7 is 7 6 x = 7 6 x + 3 .

Explanation

Identifying a Common Base We are given the equation 4 9 3 x = 34 3 2 x + 1 and asked to rewrite both sides in terms of the same base. We observe that both 49 and 343 are powers of 7.

Rewriting 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 .

Simplifying 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 .

Equating Exponents Since the bases are now equal, we can equate the exponents: 6 x = 6 x + 3 .

Solving for x and Rewriting the Equation 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 to rewrite each side of the equation in terms of base 7. So, we have 4 9 3 x = ( 7 2 ) 3 x = 7 6 x and 34 3 2 x + 1 = ( 7 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 is calculated using exponential functions. If you invest a principal amount P at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ​ ) 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.

Answered by GinnyAnswer | 2025-07-04