Are Base Ten Logarithms Relics?

Solution 1:

Yes, $10$ is not mathematically significant as a base like $e$ is. Using base 10 logs is strictly for the benefit of non-computer calculation and estimation (which, note, can include such things as simply reading a graph with a scale in dB), and consistency with previously established conventions. This may not be of interest to mathematicians, but I doubt engineers would want to give it up.

For these purposes, $10$ does have at least one useful feature beyond being the base of our number system: $\log_{10} 2 = 0.301 ≈ 0.3$. This is a very common approximation that $3$ dB corresponds to a doubling or halving of the quantity of interest. We could get similar simplicity by using $\log_{2}$, but $\log_2 10 = 3.321…$ which is not nearly as convenient for estimation in decimal numbers.

Choosing base $10$ produces nice nearly-tenth-of-an-integer results for numbers of the form $10^x2^y$ (for integer $x$ and small integer $y$), whereas an arbitrary base $b$ is only guaranteed to be nice for $b^x$.

This suggests further investigation: evaluating bases other than $10$, $2$, and $e$ for having similar almost-integer approximations. I tried writing a program to measure/plot how many good approximations there were for various bases, but it turned out that defining the goodness of an approximation and whether it's good enough to count involves a few too many parameters and I didn't get around to refining it to a result worth sharing.