What is so natural about "natural logarithms"?

Scott E. Brodie
October 8, 2007

Now what so pleasing can there be as the speculation of these things, to read and examine such experiments, or, if a man be more mathematically given, to calculate, or peruse Napier's Logarithms, or those tables of artificial sines and tangents ...

Robert Burton, The Anatomy of Melancholy
New York Review Books,
2001, p. 97 (The Second Partition)

The Oxford English Dictionary defines "logarithm" thus:

One of a particular class of arithmetical functions, invented by John Napier of Merchiston (died 1617), and tabulated for use as a means of abridging calculation. The essential property of a system of logarithms is that the sum of the logarithms of any two or more numbers is the logarithm of their product. Hence the use of a table of logarithms enables a computer to substitute addition and subtraction for the more laborious operations of multiplication and division ...

(Note the now-archaic use of the word "computer" to mean a person who performs computations!)

We can quickly review the basic facts about logarithms as follows:

Start with the simple equation for raising a number to a power:

x ^y = z.

If any two of x, y, z are known, we can determine the third:

If we know x and y, we can perform the operation of "exponentiation" and determine z.
If y and z are known, we can perform the operation of "extracting the root" to determine x:
If we know x and z, the number, y, (the exponent) which satisfies x ^y = z is called the "logarithm of z to the base x", and denoted log_x z. If the base, x, is clear from context, it need not be explicitly mentioned.

Thus, logarithms are simply exponents, and the "laws of exponents" can be paraphrased as the "laws of logarithms:"

x⁰ = 1 means log_x 1 = 0
x¹ = x means log_x x = 1
(Hence log_x y = 1 implies x = y)
x^u·x^v = x^{u + v} means log_x (a·b) = log_x a + log_x b
(x^u)^v = x^u·v means log_x a^b = b·log_x a

In particular, suppose you have a table of logarithms to the base x, but need a logarithm to the base y, say, log_y c. We have

c = x^{^{log _x c}} = y^{^{log _y c}} = (x^{^{log _x y}})^{^{log _y c}}

Taking logarithms to the base x,

log _x c = log _x y · log _y c

or,
log _y c = log _x c / log _x y.

It is thus easy to switch from one base to another, as dictated by convenience. As was first pointed out by Briggs, logarithms to the base 10 are particularly convenient for use with computations with numbers written in base 10, as is our usual practice. Often, logarithms to base 10 are referred to as "common logarithms," and are often abbreviated simply as "log x".

Of course, our affinity for writing numbers in base 10 is essentially a biological accident, an echo of the ancient practice of counting on one’s fingers, of which the most common endowment is ten. (Presumably, if arithmetic had been invented by two-toed sloths, base 4 might be in more common use!)

Euler first noticed a more universal basis for a choice of base for logarithms. He was apparently the first to contemplate the logarithm as a "function" rather than simply a table for facilitating computations. A graph of two logarithm functions is shown below. The "flatter" curve is a plot of logarithms with base 10, the "steeper" curve is a plot of logarithms to the base 2. The formula for change of base implies that any two such curves are related by a proportionality constant.

In particular, (as one might observe from the plotted curves) Euler made the simple observation, that, for values of x near x = 1, the logarithm (to any base) of x was always near zero - indeed, say, for x = 1 + y, with y a very small number, log_b x ≈ K_b·y, where the constant of proportionality, K_b, depends on the base b.

It is reassuring to note that this approximation preserves the fundamental law of logarithms:

log_b (1 + y)(1 + z)	= log_b (1 + y) + log_b (1 + z)
	= log_b (1 + y) + log_b (1 + z)
	≈ K_b·y + K_b·z
	= K_b·(y + z)
	≈ K_b·(y + z + y·z)
	≈ log_b(1 + y + z + y·z),

since y·z << y + z when y and z are very small.

Of course, as Euler recognized, the constant K_b is something of a nuisance, and it is reasonable to ask if there is a choice of base for the logarithms with which K_b = 1, that is to say, a base b for which log_b (1 + y) ≈ y, for small values of y. For this choice of base b, we have, as n increases without bound, and 1/n thus becomes arbitrarily small,

log_b (1 + 1/n) ≈ 1/n.

Multiplying through by n,

log_b (1 + 1/n)ⁿ ≈ 1.

Passing to the limit as n increases without bound, we obtain

log_b lim_n→∞(1 + 1/n)ⁿ = 1,

or,

b = lim_n→∞(1 + 1/n)ⁿ,

since the logarithm function is continuous.

Thus, lim_n→∞(1 + 1/n)ⁿ, for which Euler introduced the abbreviation "e", is the unique base b for logarithms with the property that log_b (1 + y) ≈ y for small values of y, which perhaps makes e a "natural" choice for logarithmic base, and indeed logarithms to the base e are often referred to as "natural logarithms." In books where both common and natural logarithms appear, the common logarithms are often written as "log x," and the natural logarithms are often abbreviated "ln x". (In more advanced books, where common logarithms are seldom used, the abbreviation log x is often used for natural logarithms!)

One can also easily determine the proportionality constant, K_b for other bases:

From the formula for change of base,

log_b(1 + y) = ln(1 + y) / ln b = y / ln b.

K_b = 1 / ln b.

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