Odd Base

Powering from base $b$ in both sides:

$\large b^a = b^{3/2} \cdot (\frac{11\sqrt{5}}{25} + i \frac{2\sqrt{5}}{25} ) \cdot (b^{log_be})^{i\pi}$

$\large z = b^{3/2} \cdot e^{atan(2/11)} \cdot e^{i \pi}$

Then:

$\large |z|^2 = b^3$

$\large Re(z) = -b^{3/2}\frac{11\sqrt{5}}{25}$

$\large Im(z) = -b^{3/2}\frac{2\sqrt{5}}{25}$

Summing up and multiplyting both sides by $25$ :

$\large 25b^3 - 13\sqrt{5} b^{3/2} - 2800 = 0$

Solving for $b^{3/2}$ :

$\large b^{3/2} = \frac{13 \sqrt{5} \pm 237\sqrt{5}}{50}$

$\large b^{3/2} = 5\sqrt{5}$

$\large b^3 = 125$

$\color{#3D99F6} \fbox{b=5}$