Solve the equation 2ln2x = 1 + ln3. Give your answer correct to 2dp.

To address the equation 2ln(2x) = 1 + ln(3), we begin by simplifying the left-hand side. The expression can be rewritten using the properties of logarithms, specifically the power rule, which allows us to express 2ln(2x) as ln((2x)^2). This transformation leads us to the equation ln((2x)^2) = 1 + ln(3). Next, we can isolate the logarithmic terms by manipulating the equation further.

By applying the property of logarithms that states ln(a) + ln(b) = ln(ab), we can rewrite the right-hand side of the equation. Thus, we have ln((2x)^2) = ln(3e), where e represents the base of the natural logarithm. Since the natural logarithm function is one-to-one, we can equate the arguments of the logarithms, leading us to the equation (2x)^2 = 3e. This step allows us to eliminate the logarithm and focus on solving for x.

To find the value of x, we first simplify the equation (2x)^2 = 3e. Taking the square root of both sides yields 2x = ±√(3e). Since x must be a positive value in the context of logarithms, we only consider the positive root, resulting in 2x = √(3e). Dividing both sides by 2 gives us x = √(3e)/2. Finally, substituting the approximate value of e (approximately 2.718) into the equation and calculating the result will yield the value of x, which should be rounded to two decimal places for the final answer.

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