"Exponential" growth is usually meant to be increasingly increasing; that is, it gets bigger faster and faster. Also, exponential growth generally has a fixed minimum, a horizontal line that it is never below. You can see a sampling of typical graphs at the bottom of
this page.
"Logarithmic" growth is usually meant to be decreasingly increasing; that is, it gets bigger, but slower and slower. Also, logarithmic growth generally has a restricted domain, a vertical line that it never crosses. You can see a sampling of typical graphs at the bottom of
this page.