Why is Euler's number used as a base for logarithms?

Question

Is there some special property of '$e$' which makes it suitable to be used as a base for logarithms?

Moreover, does the natural logarithm possess some advantage over the common logarithm? I don't understand why there is a need to choose an irrational number, '$e$', for a base. Isn't it much simpler to use 10 as a base?

Yes, it has many nice properties. See e.g. this or this question. — Eff, Apr 23 '16 at 13:15
"Isn't it much simpler to use 10 as a base?" - it depends on the application, of course. For things like pH and other p-values, of course the common logarithm is the convenient one. For calculus, due to the property peter mentioned (among many other things), $e$ is the "nice" base. — J. M. ain't a mathematician, Apr 23 '16 at 14:28
Interestingly, the first version of the logarithm, due to Napier, was very close in spirit to the natural logarithm. This was well before the era of Newton/Leibniz. Shortly after the work of Napier, Briggs, inspired by that work, produced tables of the base $10$ logarithm. Related tables were used for computations for centuries. — André Nicolas, Apr 23 '16 at 14:38
http://math.stackexchange.com/questions/176386/why-the-number-e-2-71828-was-chosen-as-the-natural-base-for-logarithm-function?rq=1 — MCMastery, Apr 23 '16 at 15:43
The inverse function being its own nth derivative is a cool property in my view — KR136, Apr 23 '16 at 17:12

score 31 · Accepted Answer · edited Apr 23 '16 at 13:20

31

Base 10 would be arbitrary, though it is more helpful in getting a feeling of the order of magnitude of the original numbers. The reason to pick base $e$ is that one can define $\ln$ naturally even without picking a base, for example as $$\ln x:=\int_1^x\frac{\mathrm dt}t$$ Another (or maybe the same?) advantage is that it is easy to estimate $\ln x$ for numbers close to $1$, for we have $\ln(1+h)\approx h$ if $h$ is small. Even the complete Taylor series for $\ln(1+x)$ is nice: $\ln(1+x)=x-\frac12x^2+\frac13x^3-\frac14x^4\pm\ldots$, in particular, all the coefficients are nice and rational. When working with any other base you'd have to carry an irrational constant around all the time, whereas with the natural log, this irrational constant is visible only when you write $\log_e$ instead of $\ln$.

edited Apr 23 '16 at 13:20

GoodDeeds

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answered Apr 23 '16 at 13:15

Hagen von Eitzen

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As an elementary example of the second paragraph: If I take $x=0.01$, then $\ln(x+1)\approx 0.0095\approx x$. By contrast, in base-10 we have $\log_{10}(x)\approx 0.004321$ which seems inscrutable until one notes that $\log_{10}(x)=\dfrac{\ln x}{\ln 10}\approx \dfrac{x}{\ln 10}$. – Semiclassical Apr 23 '16 at 13:43
And let's not forget Euler's formula, which gave us the first conception about the logarithm of a negative number, also used $e$. – MathIsNice1729 Apr 24 '16 at 06:10
Of course this is the same property of $e$, but the only solutions $y=ca^x$ to the simple differential equation $\frac{dy}{dx}=y$ are those which have base $a=e$. – Jeppe Stig Nielsen Apr 24 '16 at 14:52

score 6 · Answer 2 · edited Apr 13 '17 at 12:19

_{It was suggested to consider this question as a duplicate of What's so “natural” about the base of natural logarithms?. But frankly, looking at the accepted answer:}

^{If you know some linear algebra, then here is an abstract reason: $e^x$ is the unique eigenvector of eigenvalue $1$ of the derivative $D$ acting on, say, the space of smooth functions on $\mathbb{R}$.}

^{one could argue that someone who understands this answer would likely not have asked the question in the first place. ("Eigen...what?")}

For me (as a non-mathematician), the key property of this number is, geometrically speaking, that the graph of $e^x$ is describing its own slope. I tried to point this out in these images:

A little bit more formally, this means that $e$ is the only number that has the property

$$ \frac{d}{dx} e^x = e^x$$

which can be seen when plotting both functions.

Certain higher-level mathematical statements (e.g. that $e^x$ is this eigen...vector) are consequences of that.

Your last formula is exactly what was being said in your quote, but couched in admittedly very high-level language. — J. M. ain't a mathematician, Apr 24 '16 at 14:20
@J.M. Sure, I just could imagine that someone who asks this question in the first place might benefit more from an intuitive, graphical explanation than from a statement that may be hard to grok even for math graduates. At least, I liked the idea of $e$ nestling comfortably at this unique spot of $2.718...$ based on the constraint that $d/dx e^x = e^x$ - even though there are many other representations or ways of deriving the value, and many properties of the value which are important for higher-level mathematical concepts — Marco13, Apr 24 '16 at 15:19

score 2 · Answer 3 · answered Apr 24 '16 at 12:29

Historically, I believe it was because it was much easier to compute.

How do you compute $10^{4.32}$? Maybe you can raise $10^{432}$, but then how do you extract that $100^{th}$ root? Maybe you use Newton's method or something, but this is a long process, and it is not "uniform": making a table of values this way would be horrible.

However, to approximate a table of values for $e^x$ is pretty easy:

$\begin{align*} e^{0.1} &\approx \left((1+\frac{1}{10})^{10} \right)^{0.1}=1.1\\ e^{0.2} &\approx 1.1^2 = 1.21\\ e^{0.3} &\approx 1.1^3 = 1.331\\ \vdots \end{align*}$

You can improve the accuracy systematically by using finer approximations, like $e^{0.01} \approx 1.01$. All you have to do to generate the table is multiply (which for these bases amounts to shifting and adding).

Napier used $e^{0.00000001}$ to make his table of logarithms. Handling the error in the terms (which get small fast, but add up over the course of a very long table) was highly nontrivial. Once he has his log tables, it allowed a huge number of computations to be done more easily (including the creation of tools like the slide rule).

Why is Euler's number used as a base for logarithms?

3 Answers3