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To solve the equation using natural logarithms, we first isolate the exponential term by subtracting 3 from both sides:
4e^2x = 12
Then, we divide both sides by 4 to solve for e^2x:
e^2x = 3
To get rid of the natural logarithm, we take the natural logarithm (ln) of both sides:
ln(e^2x) = ln(3)
Using the property of logarithms that ln(e^a) = a, we simplify the equation to:
2x = ln(3)
Finally, divide both sides by 2 to solve for x:
x = (1/2) * ln(3)
Using a calculator, we find that x is approximately 0.549. Therefore, the answer is 0.549.