Exponential Function

For any $a>0$ and $x\in R$, $a^x$ is now fully defined (via §2.8.2's rational powers, extended by continuity to every real exponent); $1^x=1$ always. We therefore restrict attention to $a^x$ for $0<a\ne1$, and call this the exponential function with base $a$. Note $a^x$ can fail to be defined for $a<0$ (e.g. $a^{1/m}$ for even $m$), which is exactly why the base is required to be positive; also $a^x>0$ for every real $x$.

Properties of the exponential function (for $a,b>0$, $a\ne1\ne b$, all $x,y\in R$):

One-to-one and onto. For $f(x)=a^x$ ($a=2$, say): if $f(u)=f(v)$ then $2^{u-v}=1$, so $u-v=0$, i.e. $u=v$ -- $f$ is one-to-one. Since $2^0=1$, $f(x)>1$ for $x>0$ and $f(x)<1$ for $x<0$, and $f$ maps $R$ onto $(0,\infty)$. The graph of $f(x)=2^x$ increases as $x$ increases; the graph of $g(x)=(1/2)^x=1/2^x$ (base $0<a<1$) decreases as $x$ increases, with $g(0)=1$ too. In general, for any base $0<a\ne1$, $f(x)=a^x$ is one-to-one and onto, with domain $R$ and codomain (range) $(0,\infty)$ -- increasing when $a>1$, decreasing when $0<a<1$. …