Step 1: Write out the definition of sinh.
Recall that the hyperbolic sine is defined as $\sinh x = \frac{e^x - e^{-x}}{2}$. So $\sinh(2\theta) = \frac{e^{2\theta} - e^{-2\theta}}{2}$. We will substitute this into the given equation.
Step 2: Substitute into the given equation.
The equation is $3\sinh(2\theta) = 13 - 3e^{2\theta}$. Substituting: \[ 3 \cdot \frac{e^{2\theta} - e^{-2\theta}}{2} = 13 - 3e^{2\theta}. \]
Step 3: Introduce a substitution to simplify.
Let $t = e^{2\theta}$, so $e^{-2\theta} = \frac{1}{t}$ (and $t > 0$ since it is an exponential). The equation becomes: \[ \frac{3}{2}\left(t - \frac{1}{t}\right) = 13 - 3t. \]
Step 4: Clear fractions and form a quadratic.
Multiply both sides by $2t$: $3(t^2 - 1) = 26t - 6t^2$. Expanding and rearranging: \[ 3t^2 - 3 - 26t + 6t^2 = 0 \implies 9t^2 - 26t - 3 = 0. \]
Step 5: Factorise and solve the quadratic.
Factor: $9t^2 - 27t + t - 3 = 0 \implies 9t(t-3) + 1(t-3) = 0 \implies (9t+1)(t-3) = 0$. So $t = 3$ or $t = -\frac{1}{9}$. Since $t = e^{2\theta} > 0$, we discard $t = -\frac{1}{9}$.
Step 6: Solve for theta.
From $e^{2\theta} = 3$, take natural logarithm: $2\theta = \log 3$, so $\theta = \frac{1}{2}\log 3$. \[ \boxed{\theta = \dfrac{1}{2}\log 3} \]