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.
When you look at the 2 graphs side-by-side, as in the article, it's apparent which conveys more meaning (the linear one), unless I suppose you read log graphs all day. The public does not.
Your suggestion about changing what is represented in order to avoid a log scale is good but does not solve the root of the issue: many members of the public will not understand the information however it is represented because the concept itself is too complex.
It really is a numeracy problem.