Talking about plotting, in ĺog-linear plots[0], exponentials appear as straight lines. In log-log plots[1], functions of form f(x) = ax^k (monomials) appear as straight lines.
For me, I think what clicked is that the log of a number in base 10 measures the length of this number in decimal (as in, the number of digits, possibly off by one)
And more generally, log is an exponent that raises the base to a number. For example, ln(5) is the exponent that raises e to 5.
It's deeper than it sounds. Many of the data derived from physical phenomena will follow some polynomial. On a log-log plot, this looks like a squiggly but straight line, and the magic marker smooths out the lower-order terms noise :).
For me, I think what clicked is that the log of a number in base 10 measures the length of this number in decimal (as in, the number of digits, possibly off by one)
And more generally, log is an exponent that raises the base to a number. For example, ln(5) is the exponent that raises e to 5.
[0] https://en.wikipedia.org/wiki/Semi-log_plot
[1] https://en.wikipedia.org/wiki/Log%E2%80%93log_plot