# Are Base Ten Logarithms Relics?

Just interested in your thoughts regarding the contention that

the pre-eminence of base ten logarithms is a relic from pre-calculator days.

Firstly I understand that finding the (base-10) logarithm of positive real numbers without a calculator can be reduced to finding the (base-10) logarithm of numbers (strictly) between 1 and 10 via scientific notation $$\log_{10}(x)=\log_{10}(a\times 10^n)=\log_{10}(a)+\log_{10}(10^n)=\log_{10}(a)+n,\,\,\,(*)$$

and that we could compile (approximate) logarithm tables for $1<a<10$ and hence we can calculate logs base 10. However this can now be done with a calculator... and why would you want to calculate $\log_{10}(x)$ in the first place?

The next reason that we might need $\log_{10}(x)$ is to solve equations like

$$b^x=n.$$

Now we know that $x=\log_bn$ but we can use the change of base "formula" to express this in terms of log base 10. Of course the change of base "formula" comes from a calculation like

\begin{align} \log_{10}(b^x)&=\log_{10}n \\\Rightarrow x\log_{10}(b)&=\log_{10}n \\ \Rightarrow x&=\frac{\log_{10}n}{\log_{10}b}. \end{align} However the new modern calculators can calculate $\log_bn$ in the first place.

Then you could say what about solving $$b^{f(x)}=c^{g(x)}.$$ Well you don't need to take a base-10 log: we have the perfectly good base $e$ natural log!

In my presently narrow view, it seems to me that it is only stuff like the Richter Scale and sound intensity and similar derived quantities and scales that really use base-10 logs and that while base $e$ logs are clearly useful, that the pre-eminence of base 10 logs is due only to the the by-hand-calculation (*).

To ask a specific question... base $e$ is clearly special:

Are base 10 logs 'special' only because of the "ease" of calculating (or should I say approximating) logs base 10?

Or am I missing something else? The reason I am looking at this is I have a section of (precalculus) maths notes that is headed "Two Distinguished Bases" and I am thinking of throwing out base 10.

• My rather uninformed opinion is that yes, the "log_10" button on calculators is and has always been a relic mathematically. The change of base formula means that a "ln" button doesn't cost much time or space (or even readability, for what that matters on a calculator) to do the same functionality. If you're working with a calculator that has a "log_b(x)" button, well I suppose that's even more reason. On the other hand Andre has a point: "log_10" is used all the time in sciences, so if you throw out the base 10 part of the lesson, you'd better hit the change of base formula really hard. – Eric Stucky Nov 4 '13 at 21:12
• Essentially all units of physical measurement are based on powers of $10$; whether it's a "relic" or not, nobody talks about a nanometer being $e^{-20.72}$ meters. – mjqxxxx Nov 4 '13 at 21:12
• Well, there were calculators around, it once meant a person who calculates. Hard to know the answer to your question, probably yes: logs to the base $10$ "play nice" with decimal notation. Now if had $e$ fingers $\dots$. – André Nicolas Nov 4 '13 at 21:14
• Exactly $9$ more than the log base $10$ of a nanometer :). – mjqxxxx Nov 4 '13 at 21:23
• Richter Scale, pH. Lots of civil engineering stuff. – André Nicolas Nov 4 '13 at 21:38

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.

• Nice answer Kevin. – JP McCarthy Nov 4 '13 at 22:04
• Did a little work on the comparison of other bases today. Preliminary results are that 16, 2, and 4 get 'unfair' advantage due to higher density of exact multiples, and 10 comes in third place next to 3, 20, and 27. There's an awful lot of arbitrary parameters, though. – Kevin Reid Nov 7 '13 at 5:32
• @KevinReid: how did you get statistics on the prevalence of various bases? got any URL? – smci Oct 9 '15 at 0:09
• @smci No, sorry, I haven't gotten around to cleaning up the program to produce useful output. – Kevin Reid Oct 9 '15 at 1:37
• Similar to your decibel example, we have 2^10 being approximately 10^3 (a thousand). This makes estimating the conversion between base two and base ten very easy. For example, one million (10^6) is about 2^20. A billion is 2^30. A trillion is 2^40. – Paul Chernoch May 11 '16 at 16:36