Base (exponentiation)

Calculation results v t e

Addition (+) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend (broad sense)}}\,+\,{\text{addend (broad sense)}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend (strict sense)}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{sum}}}$ Subtraction (−) ${\displaystyle \scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\,=\,}$ ${\displaystyle \scriptstyle {\text{difference}}}$ Multiplication (×) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{product}}}$ Division (÷) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\text{ }}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle {\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}}$ Exponentiation ${\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}$ ${\displaystyle \scriptstyle {\text{power}}}$ nth root (√) ${\displaystyle \scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,}$ ${\displaystyle \scriptstyle {\text{root}}}$ Logarithm (log) ${\displaystyle \scriptstyle \log _{\text{base}}({\text{antilogarithm}})\,=\,}$ ${\displaystyle \scriptstyle {\text{logarithm}}}$

In exponentiation, the base is the number b in an expression of the form bⁿ.

Related terms

The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". For example, the fourth power of 10 is 10,000 because 10⁴ = 10 × 10 × 10 × 10 = 10,000. The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent.

Radix is the traditional term for base, but usually refers then to one of the common bases: decimal (10), binary (2), hexadecimal (16), or sexagesimal (60). When the concepts of variable and constant came to be distinguished, the process of exponentiation was seen to transcend the algebraic functions.

In his 1748 Introductio in analysin infinitorum, Leonhard Euler referred to "base a = 10" in an example. He referred to a as a "constant number" in an extensive consideration of the function F(z) = a^z. First z is a positive integer, then negative, then a fraction, or rational number. ^{[1] :155}

Roots

When the nth power of b equals a number a, or a = bⁿ, then b is called an "nth root" of a. For example, 10 is a fourth root of 10,000.

Logarithms

The inverse function to exponentiation with base b (when it is well-defined) is called the logarithm to base b, denoted log_b. Thus:

log_ba = n.

For example, log₁₀ 10,000 = 4.

References

1. Leonard Euler (1748) Chapter 6: Concerning Exponential and Logarithmic Quantities of Introduction to the Analysis of the Infinite, translated by Ian Bruce (2013), lk from 17centurymaths.