Question

question
solve for x, rounding to the nearest hundredth.
2e^{2x} = 97

Explanation:

Step1: Isolate the exponential term

Divide both sides of the equation \(2e^{2x} = 97\) by 2 to get \(e^{2x}=\frac{97}{2}=48.5\).

Step2: Take the natural logarithm

Take the natural logarithm of both sides: \(\ln(e^{2x})=\ln(48.5)\). Since \(\ln(e^{a}) = a\), this simplifies to \(2x=\ln(48.5)\).

Step3: Solve for x

Divide both sides by 2: \(x = \frac{\ln(48.5)}{2}\). Calculate \(\ln(48.5)\approx3.881\), then \(x=\frac{3.881}{2}\approx1.94\).

Answer:

\(x\approx1.94\)