The other day I stumbled on an article [1] that advocated writing ab as a↑b and loga(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 ab and loga(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 ex 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}

Related posts

[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

The post Alternative exp and log notation first appeared on John D. Cook.

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