Alternative exp and log notation

The other day I stumbled on an article [1] that advocated writing a^b as a↑b and log_a(b) as a↓b.

$\begin{align*} a &\uparrow b \equiv a^b \\ a &\downarrow b \equiv \log_a b \end{align*}$

This is a special case of Knuth’s up arrow and down arrow notation. Knuth introduces his arrows with the intention of repeating them to represent hyperexponentiation and iterated logarithms. But the emphasis in [1] is more on the pedagogical advantages of using a single up or down arrow.

Advantages

One advantage is that the notation is more symmetric. Exponents and logs are inverses of each other, and up and down arrows are visual inverses of each other.

Another advantage is that the down arrow notation makes the base of the logarithm more prominent, which is sometimes useful.

Finally, the up and down arrow notation is more typographically linear: a↑b and a↓b stay within a line, whereas a^b and log_a(b) extend above and below the line. LaTeX handles subscripts and superscripts well, but HTML doesn’t. That’s one reason I usually write exp(x) rather than e^x here.

Comparison

Here are the basic properties of logs and exponents using conventional notation.

$\begin{align} a^b = c &\iff \log_a c = b \\ \log_b 1 &= 0 \\ \log_b b &= 1 \\ \log_b(b^x) &= x \\ b^{\log_b x} &= x \\ \log_b xy &= \log_b x + \log_b y \\ \log_b \frac{x}{y} &= \log_b x - \log_by \\ a^{\log_b c} &= c^{\log_b a} \\ \log_a b^c &= c (\log_a b) \\ (\log_b a) (\log_a x) &= \log_b x \end{align}$

Here are the same properties using up and down arrow notation.

$\begin{align} a \uparrow b = c &\iff a \downarrow c = b \\ b \downarrow 1 &= 0 \\ b \downarrow b &= 1 \\ b \downarrow (b \uparrow x) &= x \\ b \uparrow (b \downarrow x) &= x \\ b \downarrow xy &= b \downarrow x + b \downarrow y \\ b \downarrow \frac{x}{y} &= b \downarrow x - b \downarrow y \\ a \uparrow (b \downarrow c) &= c \uparrow (b \downarrow a ) \\ a \downarrow (b \uparrow c) &= c (a \downarrow b) \\ (b \downarrow a) (a \downarrow x) &= b \downarrow x \end{align}$

[1] Margaret Brown. Some Thoughts on the Use of Computer Symbols in Mathematics. The Mathematical Gazette, Vol. 58, No. 404 (Jun., 1974), pp. 78-79

Alternative exp and log notation

Advantages

Comparison

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