Answer
(a) $b^{y}=x$
(b) $(0,\infty)$
(c) $(-\infty,\infty)$
(d) Graph is as depicted below:
Work Step by Step
(a) Since $b$ is a positive number and $b\ne1$ , the exponential function is either increasing or decreasing and so it is one-to-one by the Horizontal Line Test. It therefore has an inverse function $f^{-1}(x)$ , which is called the logarithmic function with base b and is denoted by $log_{b}x$.
$f^{-1}(x)=y$ this implies $f(y) =x$
then we have
$log_{b}x=y$ or $b^{y}=x$
(b)The logarithmic function with base b and is denoted by $f(x)=log_{b}x$ has domain $D=(0,\infty)$.
(c) The logarithmic function with base $b$ and is denoted by $f(x)=log_{b}x$ has range $R=(-\infty,\infty)$.
(d) Figure shows the case when $b > 1$. The fact that $b^{x}=y$ is a very rapidly increasing function for $x > 0$ is reflected in the fact that $log_{b}x=y$ is a very slowly increasing function for $x > 1$.The general shape of the graph of the exponential function for each of the following cases is as depicted in below figure.
