I did logarithms “by wrote” without an intuitive understanding of them until the day I realized that the logarithm base 10 is the number of digits a number is in base 10 (minus one).
log1010=1
log10100=2
log101,000=3
log1010,000=4
(But whereas “number of digits minus one” is the same for, say, 1,000 as it is for 9,999, logarithm is “smoothed”. So the logarithm of a number between 1,000 and 10,000 will be some number between 3 and 4.)
Similarly, logarithm base 2 is the number of digits in base 2 minus one. Logarithm base 16 is the number of digits in base 16 minus one. Etc.
Natural logarithm (log base Euler’s constant) is a little trickier to think in terms of, but technically it is the number of digits in base e minus one. Numerical bases that have fractional parts are a sensical concept.
I did logarithms “by wrote” without an intuitive understanding of them until the day I realized that the logarithm base 10 is the number of digits a number is in base 10 (minus one).
(But whereas “number of digits minus one” is the same for, say, 1,000 as it is for 9,999, logarithm is “smoothed”. So the logarithm of a number between 1,000 and 10,000 will be some number between 3 and 4.)
Similarly, logarithm base 2 is the number of digits in base 2 minus one. Logarithm base 16 is the number of digits in base 16 minus one. Etc.
Natural logarithm (log base Euler’s constant) is a little trickier to think in terms of, but technically it is the number of digits in base e minus one. Numerical bases that have fractional parts are a sensical concept.