Ramona's estimate of \( x \approx 2.73 \) for the equation \( 4^{x - 300} = 42 \) is unreasonable because the exponential expression \( 4^{x - 300} \) becomes dramatically smaller with negative exponents, which will occur when \( x < 300 \). Since the right side of the equation, 42, is a positive number, \( x - 300 \) must be greater than 0 to yield a positive value for \( 4^{x - 300} \). Therefore, \( x \) would need to be at least 300 for the equation to hold true, making 2.73 an insufficient value. This means Ramona's estimate falls too far below the necessary threshold to equate the two sides of the 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.
1 answer