Preheat oven to 375°F (190°C). Beat eggs and granulated sugar until light and fluffy, then gently fold in flour, cocoa powder, baking powder, and vanilla extract until just combined.

Pour the batter onto a parchment-lined 10x15 inch jelly roll pan, spreading evenly. Bake for 12-15 minutes until a toothpick inserted into the center comes out clean.

Immediately invert the warm cake onto a clean, powdered sugar-dusted kitchen towel and carefully peel off the parchment paper. Starting from a short side, roll the cake tightly with the towel inside and let it cool completely in this rolled position.

In a cold bowl, whip heavy cream, powdered sugar, and vanilla extract until stiff peaks form.

Once the cake is completely cooled, gently unroll it. Spread the whipped cream filling evenly over the cake, leaving a small border, then carefully re-roll the cake.

Beat softened butter until creamy. Gradually add powdered sugar, cocoa powder, vanilla extract, and a pinch of salt, alternating with milk/cream, until the frosting is smooth and spreadable.

Frost the entire log cake with the chocolate buttercream. Use a fork to create a wood-grain effect and decorate with desired festive garnishes like meringue mushrooms or cranberries.