The equation \(4^x - 300 = 42\) can be rearranged to \(4^x = 342\). To estimate \(x\), we can express \(4^x\) as \(2^{2x}\), which implies we are looking for \(2^{2x} = 342\). Since \(2^{10} = 1024\) and \(2^8 = 256\), we can infer that \(x\) should be slightly less than 8, since \(x\) must produce a value between these two powers of 2. Ramona's estimate of \(x \approx 2.73\) is unreasonable because it suggests \(4^{2.73} \approx 42\) which is much lower than 342. This sizable discrepancy indicates her estimate is not close to the actual solution.
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)
1 answer