One of the most common usage of logarithms that I encounter is digits. In that case I usually just write dig instead of log. Another important case when logari

One of the most common usage of logarithms that I encounter is digits. In that case I usually just write dig instead of log. Another important case when logarithm shows up, that multiplication on (0,inf) and addition on (-inf,inf) have the same structure, and logarithm creates an isomorphisms between them. Usually it makes multiplication easier/more familiar. Vi Hart has an entertaining video about it [0] where she explains how she likes to smell numbers, and visualizes the exponential function on a line written values over it, instead of a 2d graph. There are a lot of other cases where logarithm may show up. I usually treat them separately. For example if a logarithm shows up as the integral of 1/x, then chances are that I better think of it as the integral of 1/x, and I don't gain anything if I think of it as the number of digits. That doesn't happen often. [0]: https://youtu.be/N-7tcTIrers https://youtu.be/N-7tcTIrers