The second you use any of those words, people's eyes glaze over and there's an audible mumble sigh of "Ugh! Maths!"
It's sad, but it's the way it is.
Graphs should have a purpose and communicate information as clearly as possible. Scales and units should be chosen to most effectively convey information. Logarithmic scales are a poor choice for the information being displayed.
For a given set of parameters (of measures such as social distancing, mask wearing, weather, etc.) new infections will not rise by some specific, absolute number. Rather, similar environments result in similar rates of new infections as a percentage of existing, active infections.
Again graphs should serve one purpose. The purpose of those graphs is clearly, by the axis labels, supposed to display continuous deaths over a set period of time.
The linearly scaled graph displays this clearly, the logarithmic one does not. If the x-axis displayed into some future time, yes I'd agree a logarithmic scale may be more appropriate. But that would be displaying a graph of predictions at that point, which is something totally different than the two example graphs are showing.
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.
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.
As a person who creates and looks at charts all day, I can read complex charts. But I really prefer other people use linear charts of simple bars or trend lines. You're conveying a new idea to me, so don't assume I have all the context you do when reading the chart.
If you want to show acceleration, remember to use different units. Choose a unit that can be shown on a linear scale. E.g., the ratio of today's cases over the cases from a week ago.