Real Power of a Nonnegative Real Number

In the setting of The Real Numbers: Standing Notation and Background, let $\exp$ denote the exponential function and let $\log$ denote the natural logarithm, which is defined at every positive real number.

1. (Power with nonnegative base)¶ Let $a$ be a positive real number. For a nonnegative real number $t$, the power $t^{a}$ of $t$ with exponent $a$ is the real number defined by

A nonnegative real number is positive or equal to $0$, and not both, so exactly one of the two clauses applies to each $t$; in the first clause $\log t$ is defined because $t>0$. Hence $t^{a}$ is a well-defined real number for every nonnegative $t$ and every positive $a$.

For $t>0$ the value $\exp(a\log t)$ is by definition the real power of a positive real number with exponent $a$, so the notation $t^{a}$ is unambiguous and the two readings may be used interchangeably wherever both apply. The present item extends that notation to the base $0$, and only for positive exponents: no meaning is assigned here to $0^{a}$ with $a\le 0$.