Superman 4 (Spring 1940) is the first issue of the comic to actually be what the series is renowned for being: original stories all featuring one character. There are four tales in the issue, the first two of which feature Luthor.
Siegel, Shuster and Cassidy had already introduced Luthor in Action 23, although in reality this book came out just before that one. But these Luthor stories clearly take place after the one in Action Comics, as Superman knows who Luthor is. It has long been conjectured that the bald assistant, talking to Luthor on the viewscreen in the page above, was mistaken by a later artist for Luthor, which is why the character went from having red hair to none.
In this story, aside from making his face appear in a tree, Luthor gets ahold of a deadly new invention, a machine that creates earthquakes. He challenges Superman to a series of tests of superiority, while his henchmen get the device.
The tests do nicely illustrate the rivalry between the characters, Luthor’s genius against Superman’s physical superiority. Superman also destroys the earthquake machine before Luthor can use, but the villain himself escapes.
Only to return a couple of pages later, in the second story in the issue. Once again Luthor demonstrates his ability to project his face onto objects, in this case a rocket ship.
Luthor has his men kidnap Clark Kent and Lois Lane, after the trouble they caused him in Europe, in his first appearance. Luthor has raised Pacifiq, a sunken city of technological marvels and scary ancient beasts.
Luthor appears to fall victim to the monstrous creatures himself, after Superman destroys his machines and rescues Lois, and Pacifiq sinks back beneath the waves.
I should also mention that, as of this story, Lois and Clark work for the Daily Planet, no longer the Daily Star, although still under George Taylor as editor.
Luthor returns in the next issue.