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.

Answer 1

Ramona's estimated solution of \( x \approx 2.73 \) is unreasonable because substituting this value into the left side of the equation \( 4^{x} - 300 = 42 \) leads to a left-hand side value that is much larger than 42. For instance, calculating \( 4^{2.73} \) yields approximately 21.5, which means \( 4^{2.73} - 300 \) is a significantly negative number, clearly not equal to 42. Additionally, exponentiating 4 by 2.73 results in a value that does not support the equation, indicating that a value of \( x \) less than 4 is needed to balance the equation correctly. Hence, her estimate does not satisfy the conditions set by the original equation.