Question

solve: \ln 2x + \ln 2 = 0
x =
done

Explanation:

Step1: Use logarithm property

Using the property of logarithms \(\ln a+\ln b = \ln(ab)\), we can combine the left - hand side of the equation \(\ln 2x+\ln 2 = 0\). So we get \(\ln((2x)\times2)=\ln(4x)=0\).

Step2: Convert logarithmic to exponential form

Recall that if \(\ln y = k\), then \(y = e^{k}\) (since the natural logarithm \(\ln\) has a base of \(e\)). For the equation \(\ln(4x)=0\), we can rewrite it in exponential form. Here, \(k = 0\) and \(y = 4x\), so \(4x=e^{0}\).

Step3: Simplify and solve for x

Since \(e^{0}=1\), the equation \(4x = e^{0}\) becomes \(4x = 1\). Then, to solve for \(x\), we divide both sides of the equation by 4: \(x=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\)