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 February 12th, 2019, 02:59 AM #1 Newbie   Joined: Feb 2019 From: home Posts: 5 Thanks: 0 mathematics help If a function $f$ is inverse of another function $g$, then domain and codomain of $f$ are codomain and domain of $g$ respectively. My book has written that exponential function $a^x$ has codomain $R$, but since logarithmic function is its inverse then why logarithmic function's domain doesn't matches with codomain of $a^x$? Last edited by skipjack; February 12th, 2019 at 01:23 PM.
 February 12th, 2019, 04:30 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond There are exceptions to that rule. Consider, for example, $y=x^2$. Thanks from Denis and topsquark
 February 12th, 2019, 12:42 PM #3 Global Moderator   Joined: May 2007 Posts: 6,823 Thanks: 723 You are better off using range rather than codomain. Thanks from topsquark
February 12th, 2019, 03:34 PM   #4
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Quote:
 Originally Posted by suyogya If a function $f$ is inverse of another function $g$, then domain and codomain of $f$ are codomain and domain of $g$ respectively. My book has written that exponential function $a^x$ has codomain $R$, but since logarithmic function is its inverse then why logarithmic function's domain doesn't matches with codomain of $a^x$?
This is wrong. The correct statement is the following:

If $f,g$ are inverses, then the domain of $f$ is the image of $g$ and the domain of $g$ is the image of $f$.

With the corrected version it is easily checked for your example. The (maximum) domain for $\log(x)$ is $(0,\infty)$ and as expected, the image of $e^x$ on the domain $\mathbb{R}$ is exactly $(0,\infty)$.

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