Question

Ramona was asked to estimate the solution to the exponential equation 4x−300=42 by using an over/under table and providing her answer to two decimal places. She gave an estimated solution of x≈2.73 . In 3–5 sentences, explain why this is an unreasonable estimate of the solution to this equation.(4 points)

Answer 1

Ramona's estimate of \( x \approx 2.73 \) is unreasonable because when we substitute this value into the equation \( 4^{x} - 300 = 42 \), it results in calculating \( 4^{2.73} \). This value is much larger than 342 (since \( 4^{2.73} \approx 387.67 \)), leading to the left side being significantly greater than 42, which means \( 4^{x} \) is far from the value of 342 we need to satisfy the equation. Therefore, a more accurate estimate would need to be lower to ensure that \( 4^{x} \) approaches 342. This discrepancy indicates that her estimate is not valid within the context of solving the exponential equation.