Exponential Function

7. Exponential function. Form: $f(x)=a^x$ is an exponential function with base $a$ and exponent (or index) $x$, where $a\ne0$, $a>0$, and $x\in R$. Example: $f(x)=2^x$ and $f(x)=2^{-x}$ (Fig. 6.28). Domain: $R$; Range: $(0,\infty)$.

Properties: (1) As $x\to-\infty$, $f(x)=2^x\to0$, so the graph has horizontal asymptote $y=0$. (2) Taking the natural base $e$ ($\approx2.718$), the graph of $f(x)=e^x$ (Fig. 6.29) has a similar appearance to $2^x$. (3) For $a>0,\ a\ne1$: if $a^x=a^y$ then $x=y$ — so $a^x$ is a one-one function (check the graph against the horizontal line test). (4) $r>1,\ m>n\implies r^m>r^n$; and $r<1,\ m>n\implies r^m<r^n$.

Ex. 9: Solve $5^{2x+7}=125$.

Solution: $5^{2x+7}=125$, i.e. $5^{2x+7}=5^3$, so $2x+7=3$, and $x=\dfrac{3-7}{2}=\dfrac{-4}{2}=-2$.

Ex. 10: Find the domain of $f(x)=\sqrt{6-2^x-2^{3-x}}$. …