Questions: Solve the following equation for x. 124 = 4 e^(4x-2) Note: Round your answer to three decimal digits (the nearest thousandth).

Solution Steps

To solve the equation \(124 = 4e^{4x-2}\) for \(x\), follow these steps:

1. Divide both sides of the equation by 4 to isolate the exponential term.
2. Take the natural logarithm (ln) of both sides to solve for the exponent.
3. Rearrange the equation to solve for \(x\).
4. Round the result to three decimal places.

Starting with the equation: \[ 124 = 4e^{4x - 2} \] we divide both sides by 4: \[ 31 = e^{4x - 2} \]

Next, we take the natural logarithm of both sides: \[ \ln(31) = 4x - 2 \]

Rearranging the equation to solve for \(x\): \[ 4x = \ln(31) + 2 \] \[ x = \frac{\ln(31) + 2}{4} \] Calculating the value gives: \[ x \approx \frac{3.4339872044851463 + 2}{4} \approx 1.3584968011212866 \] Rounding to three decimal places, we find: \[ x \approx 1.358 \]

Final Answer