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 xapprox 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 \( x \approx 2.73 \) seems unreasonable because, when plugging this value back into the exponential equation, \( 4^{2.73} \) significantly exceeds 300. Specifically, \( 4^{2.73} \) is approximately 22.63, which means \( 4^{2.73} - 300 \) would be far less than 42, failing to satisfy the equation \( 4^x - 300 = 42 \). Given that the equation can be rewritten as \( 4^x = 342 \), the correct value of \( x \) must be higher than Ramona's estimate, suggesting a closer examination of the calculations or the use of a more precise method, such as logarithmic solving, would be beneficial for accuracy. Therefore, this estimate does not align with the exponential growth pattern of the function at play.