A well-known formula for the logarithm is given by

$$\log x = -\frac{\pi}{AGM(a^2,b^2)}, \qquad x < 1$$

where AGM is the arithmetic-geometric mean, and $a$ and $b$ are given by

$$a = \sum_{k\in\mathbb{Z}}x^{k^2}, \qquad b = \sum_{k\in\mathbb{Z}}x^{(k+1/2)^2};$$

Or, equivalently,

$$a = 1 + 2x + 2x^4 + 2x^9 + 2x^{16} + 2x^{25} + \dots$$ and $$b = 2x^{1/4} + 2x^{9/4} + 2x^{25/4} + 2x^{49/4} + \dots$$

Why does it happen that $a = b$ occurs when $x = 0.8$?