Start by taking the natural logarithm of both sides of the equation:

\(\ln(4^{3x-1}) = \ln(3 \cdot 5^x)\)

Apply the logarithm power rule: \(\ln(a^b) = b \ln(a)\).

\((3x-1) \ln 4 = \ln 3 + x \ln 5\)

Expand and rearrange the equation to solve for \(x\):

\(3x \ln 4 - \ln 4 = \ln 3 + x \ln 5\)

\(3x \ln 4 - x \ln 5 = \ln 3 + \ln 4\)

Factor out \(x\):

\(x(3 \ln 4 - \ln 5) = \ln 3 + \ln 4\)

Solve for \(x\):

\(x = \frac{\ln 3 + \ln 4}{3 \ln 4 - \ln 5}\)

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

\(x \approx 0.975\)

Take the natural logarithm of both sides of the equation:

\(\ln(2^{5x}) = \ln(3^{2x+1})\)

Apply the power rule for logarithms: \(a \ln(b) = \ln(b^a)\).

\(5x \ln(2) = (2x+1) \ln(3)\)

Expand the right side:

\(5x \ln(2) = 2x \ln(3) + \ln(3)\)

Rearrange to solve for \(x\):

\(5x \ln(2) - 2x \ln(3) = \ln(3)\)

Factor out \(x\):

\(x (5 \ln(2) - 2 \ln(3)) = \ln(3)\)

Solve for \(x\):

\(x = \frac{\ln(3)}{5 \ln(2) - 2 \ln(3)}\)

Calculate \(x\) to 3 significant figures:

\(x \approx 0.866\)

Start with the equation \(5^{2x-1} = 2(3^x)\).

Take the natural logarithm of both sides: \(\ln(5^{2x-1}) = \ln(2 \cdot 3^x)\).

Apply the logarithm power rule: \((2x-1)\ln 5 = \ln 2 + x \ln 3\).

Rearrange to form a linear equation: \(2x \ln 5 - x \ln 3 = \ln 2 + \ln 5\).

Factor out \(x\): \(x(2\ln 5 - \ln 3) = \ln 2 + \ln 5\).

Solve for \(x\): \(x = \frac{\ln 2 + \ln 5}{2\ln 5 - \ln 3}\).

Calculate \(x\) to 3 significant figures: \(x = 1.09\).