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 if we substitute this value into the equation \( 4^x - 300 = 42 \), we can see that \( 4^{2.73} \) is likely much greater than 342 (since \( 4^x \) grows rapidly). Specifically, \( 4^{2.73} \) approximates to about 16.8, which means \( 4^{2.73} - 300 \) would be negative, suggesting that her estimate does not satisfy the equation. A more reasonable approach would involve checking values around \( x = 2 \) or \( x = 3 \) to find where \( 4^x \) approaches 342, which would yield a more accurate estimate for \( x \).