Ramona's estimate of \( x \approx 2.73 \) for the equation \( 4^x - 300 = 42 \) is unreasonable because substituting this value into the equation results in a significantly higher output. Calculating \( 4^{2.73} \) gives approximately \( 23.28 \), thus \( 4^{2.73} - 300 \) would yield a very negative value, indicating that it is much less than 42. Therefore, Ramona's estimation does not meet the requirement that \( 4^x \) must equal 342 (since \( 4^x - 300 = 42 \)), reflecting that her estimate is too low. A more accurate range should be determined by recalculating \( 4^x \) for values closer to the target of 342.
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.(4 points)
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