Properties of Real Powers

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $t^{a}$ be the real power of Real Powers (i) and Real Powers (ii). Let $\log$ be the natural logarithm, let $[0,\infty)$ and $(0,\infty)$ be the rays of nonnegative and of positive reals, and let $a,b,s,t\in\mathbb{R}$.

(i) Natural exponents. Let $t\ge0$ and $n\in\mathbb{N}_{0}$, with $n\ge1$ if $t=0$. Then the real power $t^{n}$, with $n$ read as a real number, equals the power $t^{n}$ of Powers with Exponents in the Natural Numbers with Zero. In particular $t^{1}=t$, and $t^{0}=1$ if $t>0$.

(ii) Sign. If $t>0$, then $t^{a}>0$. If $t\ge0$ and $a>0$, then $t^{a}\ge0$, and $t^{a}=0$ if and only if $t=0$.

(iii) Logarithm. If $t>0$, then

$$\log t^{a}=a\log t .$$

(iv) Exponent laws. If $t>0$, then

$$t^{a+b}=t^{a}t^{b},\qquad (t^{a})^{b}=t^{ab},\qquad t^{-a}=\frac{1}{t^{a}} .$$

If $t\ge0$ and $a,b>0$, then $t^{a+b}=t^{a}t^{b}$ and $(t^{a})^{b}=t^{ab}$.

(v) Products and quotients. If $s,t>0$, then

$$(st)^{a}=s^{a}t^{a},\qquad \Bigl(\frac{s}{t}\Bigr)^{a}=\frac{s^{a}}{t^{a}} .$$

If $a>0$, the first identity holds for all $s,t\ge0$, and the second for all $s\ge0$ and $t>0$.

(vi) Monotonicity. If $a>0$ and $0\le s<t$, then $s^{a}<t^{a}$.

(vii) Inverse. If $a>0$, the function $P_{a}:[0,\infty)\to[0,\infty)$, $t\mapsto t^{a}$, is bijective, and its inverse is $P_{1/a}$:

$$(t^{a})^{1/a}=t=(t^{1/a})^{a}\qquad(t\ge0).$$

(viii) Continuity. If $a>0$, the function $[0,\infty)\to\mathbb{R}$, $t\mapsto t^{a}$, is continuous on $[0,\infty)$, as in The Real Line: Standing Notation and Background for Calculus on Intervals (vi), and the function $\mathbb{R}\to\mathbb{R}$, $u\mapsto|u|^{a}$, is continuous on $\mathbb{R}$.