Start with the equation \(\ln(e^{2x} + 3) = 2x + \ln 3\).

Use the property of logarithms: \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).

Rearrange the equation: \(\ln(e^{2x} + 3) - \ln 3 = 2x\).

This becomes \(\ln\left(\frac{e^{2x} + 3}{3}\right) = 2x\).

Exponentiate both sides to remove the logarithm: \(\frac{e^{2x} + 3}{3} = e^{2x}\).

Multiply through by 3: \(e^{2x} + 3 = 3e^{2x}\).

Rearrange to form a quadratic in terms of \(e^{2x}\): \(3 = 2e^{2x}\).

Divide by 2: \(e^{2x} = \frac{3}{2}\).

Take the natural logarithm of both sides: \(2x = \ln\left(\frac{3}{2}\right)\).

Solve for \(x\): \(x = \frac{1}{2} \ln\left(\frac{3}{2}\right)\).

Calculate \(x\) to 3 decimal places: \(x \approx 0.203\).

Start with the equation \(2(3^{2x-1}) = 4^{x+1}\).

Take the logarithm of both sides:

\(\ln(2) + (2x-1)\ln(3) = (x+1)\ln(4)\).

Rearrange to form a linear equation:

\((2x-1)\ln(3) - x\ln(4) = \ln(4) - \ln(2)\).

Simplify and solve for \(x\):

\(x = \frac{\ln(4) - \ln(2) + \ln(3)}{2\ln(3) - \ln(4)}\).

Calculate the value of \(x\) to 2 decimal places:

\(x \approx 2.21\).

Start with the equation \(3(2^{1-x}) = 7^x\).

Take the natural logarithm of both sides: \(\ln(3(2^{1-x})) = \ln(7^x)\).

Using the logarithm properties, this becomes \(\ln 3 + (1-x)\ln 2 = x\ln 7\).

Rearrange to form a linear equation: \(\ln 3 + \ln 2 - x\ln 2 = x\ln 7\).

Combine terms: \(\ln 3 + \ln 2 = x\ln 7 + x\ln 2\).

Factor out \(x\): \(\ln 3 + \ln 2 = x(\ln 7 + \ln 2)\).

Solve for \(x\): \(x = \frac{\ln 3 + \ln 2}{\ln 7 + \ln 2}\).

Simplify: \(x = \frac{\ln 6}{\ln 14}\).

Start with the given equation:

\(\ln(x^3 - 3) = 3 \ln x - \ln 3\)

Apply the logarithm property \(a \ln b = \ln(b^a)\):

\(\ln(x^3 - 3) = \ln(x^3) - \ln 3\)

Use the property \(\ln a - \ln b = \ln\left(\frac{a}{b}\right)\):

\(\ln(x^3 - 3) = \ln\left(\frac{x^3}{3}\right)\)

Since the logarithms are equal, set the arguments equal:

\(x^3 - 3 = \frac{x^3}{3}\)

Multiply through by 3 to clear the fraction:

\(3(x^3 - 3) = x^3\)

\(3x^3 - 9 = x^3\)

Rearrange to solve for \(x^3\):

\(3x^3 - x^3 = 9\)

\(2x^3 = 9\)

Divide by 2:

\(x^3 = \frac{9}{2}\)

Take the cube root:

\(x = \sqrt[3]{\frac{9}{2}}\)

Calculate \(x\) to 3 significant figures:

\(x \approx 1.65\)

Let \(y = e^{2x}\). Then \(e^{-2x} = \frac{1}{y}\).

The equation becomes:

\(3y - \frac{4}{y} = 5\).

Multiply through by \(y\) to clear the fraction:

\(3y^2 - 5y - 4 = 0\).

This is a quadratic equation in \(y\). Solve using the quadratic formula:

\(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = -5\), \(c = -4\).

\(y = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3}\)

\(y = \frac{5 \pm \sqrt{25 + 48}}{6}\)

\(y = \frac{5 \pm \sqrt{73}}{6}\)

Since \(y = e^{2x}\), we have:

\(e^{2x} = \frac{5 + \sqrt{73}}{6}\)

Take the natural logarithm of both sides:

\(2x = \ln\left(\frac{5 + \sqrt{73}}{6}\right)\)

\(x = \frac{1}{2} \ln\left(\frac{5 + \sqrt{73}}{6}\right)\)

Calculating this gives:

\(x \approx 0.407\)