Step-by-Step Solution

1. Given the differential equation: \(\log(\frac{dy}{dx}) = 3x + 4y\)

2. Exponentiate both sides to remove the logarithm: \(\frac{dy}{dx} = e^{3x + 4y}\)

3. Rewrite the exponential term: \(\frac{dy}{dx} = e^{3x} \cdot e^{4y}\)

4. Separate the variables: \(e^{-4y} dy = e^{3x} dx\)

5. Integrate both sides: \(\int e^{-4y} dy = \int e^{3x} dx\)

6. Perform the integration: \(-\frac{1}{4} e^{-4y} = \frac{1}{3} e^{3x} + C\)

7. Multiply both sides by -12 to eliminate fractions: \(3e^{-4y} = -4e^{3x} - 12C\)

8. Rearrange the equation: \(3e^{-4y} + 4e^{3x} + 12C = 0\)