Real Powers

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\exp:\mathbb{R}\to\mathbb{R}$ be the exponential function and let $\log:(0,\infty)\to\mathbb{R}$ be the natural logarithm, defined on the ray $(0,\infty)$ of positive reals.

(i) Positive base. For real numbers $t>0$ and $a$, the power of $t$ with exponent $a$ is the real number

$$t^{a}=\exp(a\log t).$$

(ii) Zero base. For a real number $a>0$, the power of $0$ with exponent $a$ is

$$0^{a}=0.$$