Question

Ramona was asked to estimate the solution to the exponential equation 4^x-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.

Answer 1

Ramona's estimate of \( x \approx 2.73 \) is unreasonable because substituting this value into the left side of the equation gives \( 4^{2.73} - 300 \) which results in a number much larger than 42. Specifically, \( 4^{2.73} \) is approximately 28.54 (since \( 4^2 = 16 \) and \( 4^3 = 64 \)), making \( 4^{2.73} - 300 \approx -271.46 \), far from the target of 42. Thus, her estimate does not accurately satisfy the equation, indicating the need for a lower \( x \) value to achieve a more reasonable approximation. This inconsistency suggests that a recalibration of her estimate is necessary.