1. **State the problem:** We have an isosceles triangle with two equal sides each of length $\sqrt{52}$ and a base of length 8. The height $x$ is drawn from the apex to the midpoint of the base. We need to find the value of $x$.\n\n2. **Analyze the triangle:** The height $x$ splits the base into two equal segments of length $\frac{8}{2} = 4$. It also forms two right triangles with hypotenuse $\sqrt{52}$, one leg $x$ (the height), and the other leg 4 (half the base).\n\n3. **Apply the Pythagorean theorem:** For one of the right triangles, we have $$x^2 + 4^2 = (\sqrt{52})^2.$$\n\n4. **Simplify the equation:** $$x^2 + 16 = 52.$$\n\n5. **Solve for $x^2$:** $$x^2 = 52 - 16 = 36.$$\n\n6. **Find $x$:** $$x = \sqrt{36} = 6.$$\n\n**Final answer:** The height $x$ is 6.