Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.4* General Logarithmic and Exponential Functions - 6.4* Exercise: 2

Answer

(a) $b^{y}=x$ (b) $(0,\infty)$ (c) $(-\infty,\infty)$ (d) Graph is as depicted below:
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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.
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