In the following, $x$ and $y$ are arbitrary real numbers, and $q$ is an arbitrary constant that is strictly bigger than zero.
• $q^0=1$
• $q^{x+y}=q^xq^y\text{,}$ $q^{x-y}=\frac{q^x}{q^y}$
• $q^{-x}=\frac{1}{q^x}$
• $\big(q^x\big)^y=q^{xy}$
• $\lim\limits_{x\rightarrow\infty}q^x=\infty\text{,}$ $\lim\limits_{x\rightarrow-\infty}q^x=0$ if $q \gt 1$
• $\lim\limits_{x\rightarrow\infty}q^x=0\text{,}$ $\lim\limits_{x\rightarrow-\infty}q^x=\infty$ if $0 \lt q \lt 1$
• The graph of $2^x$ is given below. The graph of $q^x\text{,}$ for any $q \gt 1\text{,}$ is similar.