Ramona's estimate of \( x \approx 2.73 \) is unreasonable because, when substituted back into the equation \( 4^{x} - 300 = 42 \), the left-hand side does not yield a result close to 42. In fact, calculating \( 4^{2.73} \) results in a value significantly greater than 300, leading to a left-hand side far exceeding the right-hand side. To accurately solve the equation, the value of \( 4^{x} \) should be within a range that makes \( 4^{x} - 300 \) approximately equal to 42, which would imply \( 4^{x} \) should be around 342, suggesting that \( x \) must be lower than 2.73. Thus, her estimate fails to satisfy the exponential equation properly.
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)
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