Etymology of 'exponent' and 'logarithm'

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Exponents

In mathematics, “exponent” is the primary English term for 指数.

You can break it down for memorization:

exponent = ex- + pon/pos + -ent

  • ex-: “outward,” “out”
  • pon/pos: “to place,” “to put,” sharing a root with position, compose, and deposit
  • Thus, exponent originally meant “the person or thing that places something out or makes it explicit.”

In mathematics, think of it as: the exponent is the number that is “placed outside/above” the base.

For example:

2^3
  • 2 = base (底数)
  • 3 = exponent (指数)
  • 2^3 = power (幂 / result of exponentiation)

Pronunciation:

2^3 = two to the third power

Related terms:

  • exponent: exponent—the superscript number
  • exponential: exponential; pertaining to exponential functions
    e.g., exponential growth = exponential growth
  • exponential function: exponential function, e.g., y = 2^x

Note: In Chinese, “指数” may also mean “stock index,” “price index,” etc.—in such contexts, the English term is index, not exponent.

Logarithms

logarithm can be broken down as:

log- + arithm-

  • log-: from Greek logos, here meaning “ratio,” “relation,” or “computation.”
  • arithm-: from Greek arithmos, meaning “number”; cognate with arithmetic.

Thus, logarithm originally meant: “a number expressing a ratio or relationship,” which aligns with the Chinese term “对数.”

Memory aid:

logarithm = log + arithmetic

  • Seeing arithm should remind you of arithmetic—both relate to “numbers” and “computation.”
  • Seeing log should remind you of the mathematical log, which asks:
a^x = b  
log_a b = x

That is: log asks, “What is the exponent x?”

For example:

2^3 = 8  
log_2 8 = 3

One-sentence summary: A logarithm is “something that converts multiplicative relationships into exponential numbers.”

Caution: Don’t confuse it with algorithm:

  • logarithm = logarithm—related to log + arithmetic
  • algorithm = algorithm—derived from the name of the mathematician Al-Khwarizmi; not from the same root