How do I test a jest console.log

If you want to check that console.log received the right parameter (the one that you passed in) you should check mock of your jest.fn().
You also have to invoke your log function, otherwise console.log is never invoked:

it('console.log the text "hello"', () => {
  console.log = jest.fn();
  log('hello');
  // The first argument of the first call to the function was 'hello'
  expect(console.log.mock.calls[0][0]).toBe('hello');
});

or

it('console.log the text "hello"', () => {
  console.log = jest.fn();
  log('hello');
  // The first argument of the first call to the function was 'hello'
  expect(console.log).toHaveBeenCalledWith('hello');
});

If you're going with this approach don't forget to restore the original value of console.log.

Another option is to use jest.spyOn (instead of replacing the console.log it will create a proxy to it):

it('console.log the text "hello"', () => {
  const logSpy = jest.spyOn(console, 'log');

  console.log('hello');

  expect(logSpy).toHaveBeenCalledWith('hello');
});

Read more here.

Or you could do it like this:

it('calls console.log with "hello"', () => {
  const consoleSpy = jest.spyOn(console, 'log');

  console.log('hello');

  expect(consoleSpy).toHaveBeenCalledWith('hello');
});