Logarithms may be the least understood topic in basic math. In my experience, if an otherwise math-savvy person is missing something elementary, it’s usually logarithms.
For example, I have had conversations with people with advanced technical degrees where I’ve had to explain that logs in all bases are proportional to each other. For example, if one thing is proportional to the natural log of another, the former is also proportional to the log base 10 or log base anything else of the latter .
I’ve also noticed that quite often there’s a question on the front page of math.stackexchange of the form “How do I solve …” and the solution is invariably “take logarithms of both sides.” This seems to be a secret technique.
I suspect that more people understood logarithms when they had to use slide rules. A slide rule is essentially two sticks with log-scale markings. By moving one relative to the other, you’re adding lengths, which means adding logs, which does multiplication. If you do that for a while, it seems you’d have to get a feel for logs.
Log tables also make logs more tangible. At first it seems there’s no skill required to use a table, but you often have to exercise a little bit of understanding. Because of the limitations of space, tables can’t be big enough to let you directly look up everything. You have to learn how to handle orders of magnitude and how to interpolate.
If the first time you see logs is when it’s time to learn to differentiate them, you have to learn two things at once. And that’s too much for many students. They make mistakes, such as assuming logs are linear functions, that they would not make if they had an intuitive feel for what they’re working with.
Maybe schools could have a retro math week each year when students can’t use calculators and have to use log tables and slide rules. I don’t think it would do as much good to just make tables or slide rules a topic in the curriculum. It’s when you have to use these things to accomplish something else, when they are not simply an isolated forgettable topic of their own, that the ideas sink in.
More on logarithms
 That is, loga(x) = loga(b) logb(x). This says loga(b) is the proportionality constant for converting between logs in base a and b. To prove the equation, raise a to the power of both sides.
To memorize this equation, notice the up-and-down pattern of the bases and arguments: a up to x = a up to b down to b up to x. The right side squeezes an up and down b in between a and x.