boost vs leg up

boost

noun
  • Something that helps, or adds power or effectiveness; assistance. 

  • A positive intake manifold pressure in cars with turbochargers or superchargers. 

  • A coordinate transformation that changes velocity. 

  • A push from behind, as to one who is endeavoring to climb. 

verb
  • To lift or push from behind (one who is endeavoring to climb); to push up. 

  • To steal. 

  • To give a booster shot to. 

  • To help or encourage (something) to increase or improve; to assist in overcoming obstacles. 

  • To amplify; to signal boost. 

  • To jump-start a vehicle by using cables to connect the battery in a running vehicle to the battery in a vehicle that won't start. 

leg up

noun
  • Assistance, help given to get started or overcome a problem or obstacle. 

  • A boost, a sudden improvement; an advantage, an edge. 

  • The act of assisting another's progress over a wall or other obstacle by forming a step for one of their feet with one's hands. 

How often have the words boost and leg up occurred in a corpus of books? (source: Google Ngram Viewer )