I've found that, if somebody has trouble understanding a type of statistic (log-scaled, survival rate, confidence interval), it's because they never use them. You could say we should teach these things in high school, but why? So we can effectively communicate data they still won't use?
Throwing more math, tech notes, or gentle introductions at the communication problem has never worked for me. What has worked is expressing the concept in the most "real" terms possible.
Epidemiologists are worried about exponential growth because it means hospitals will be swamped. That worry should be the topic when communicating with the public. How long until that's projected to happen? Maybe show the linear trend with a horizontal line for capacity and dotted line for a forecast of need. Put a huge red circle around where they intersect. Even better: just write "X days predicted until overrun."
Also, charts are terrible for reading data. They're good for comparisons, trends, and distributions. I only use them for exploring or summarizing. Tables are for readable numbers.