Start by dividing both sides of the equation by 5:

\(\ln(4 - 3^x) = \frac{6}{5}\).

Exponentiate both sides to remove the logarithm:

\(4 - 3^x = e^{\frac{6}{5}}\).

Calculate \(e^{\frac{6}{5}}\) to get approximately 3.3201169.

Thus, \(4 - 3^x = 3.3201169\).

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

\(3^x = 4 - 3.3201169\).

\(3^x = 0.6798831\).

Take the logarithm base 3 of both sides:

\(x = \log_3(0.6798831)\).

Calculate \(x\) to get approximately \(-0.351\).