Select the correct answer.
Solve for $x$ in this equation.
$7^{3 x-17}=49^x$
A. 2
B. 15
C. 17
D. 20
Answer (2)
To solve the equation 7 3 x − 17 = 4 9 x , we rewrite it with a common base as 7 3 x − 17 = 7 2 x and equate the exponents, leading to x = 17 . Verification shows both sides are equal. The correct answer is 17 .
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17
Explanation
Rewrite the equation We are given the equation 7 3 x − 17 = 4 9 x . Our goal is to solve for x . We can start by rewriting the equation such that both sides have the same base. Since 49 = 7 2 , we can rewrite the equation as 7 3 x − 17 = ( 7 2 ) x .
Simplify the equation Using the power of a power rule, we have ( 7 2 ) x = 7 2 x . So our equation becomes 7 3 x − 17 = 7 2 x .
Equate the exponents Since the bases are equal, the exponents must be equal. Therefore, we have the equation 3 x − 17 = 2 x .
Solve for x Now, we solve the linear equation for x . Subtract 2 x from both sides of the equation: 3 x − 2 x − 17 = 2 x − 2 x , which simplifies to x − 17 = 0 .
Final solution Add 17 to both sides of the equation: x − 17 + 17 = 0 + 17 , which gives us x = 17 .
Verification To verify the solution, substitute x = 17 into the original equation: 7 3 ( 17 ) − 17 = 7 51 − 17 = 7 34 and 4 9 17 = ( 7 2 ) 17 = 7 34 . Since both sides are equal, the solution is correct.
Examples
Imagine you're baking a cake and need to adjust the oven temperature. This problem demonstrates how exponential equations can be used to model growth or decay. For instance, understanding how temperature changes over time (exponential growth or decay) helps you set the oven correctly to achieve the desired baking result. Similarly, in finance, understanding exponential growth helps calculate investments over time, ensuring you reach your financial goals.
