Prove basic properties of the exponential function

A real number $a$ raised to a real exponent $x$ is defined by

$a^x = e^{x \log a}.$

Prove the following properties:

$a^x = e^{x \log a}$

$e^x$

$E(x)$

$e^x = E(x)$

Proof.
$\log a^x = \log (e^{x \log a}) = x \log a \log e = x \log a. \qquad \blacksquare$
Proof.
$(ab)^x = e^{x \log (ab)} = e^{x \log a + x \log b} = e^{x \log a}e^{x \log b} = a^x b^x. \qquad \blacksquare$
Proof.
$a^x a^y = (e ^{x \log a})(e^{y \log a}) = e^{x \log a + y \log a} = e^{(x+y)\log a} = a^{x+y}. \qquad \blacksquare$
Proof.
$\begin{align*} (a^x)^y &= e^{y \log a^x} = e^{xy \log a} = a^{xy} \\ (a^y)^x &= e^{x \log a^y} = e^{xy \log a} = a^{xy}. \qquad \blacksquare \end{align*}$
Proof. Assume $a \neq 1$ . For the forward direction, assume $y = a^x$ . We have
$y = a^x = e^{x \log a} \quad \implies \quad \log y = \log \left(e^{ x \log a} \right) = x \log a.$

Therefore, since $a \neq 1$ we have $\log a \neq 0$ , and so

$x = \frac{\log y}{\log a} \quad \implies \quad x = \log_a y.$

Conversely, if $x = \log_a y$ then we have

$x \log a = log y \quad \implies \quad e^{x \log a} = e^{\log y} \quad \implies \quad a^x = y. \qquad \blacksquare$