Mathematica functions for antilogarithm[b, x] and cologarithm[b, x] don't appear to exist, but I think they could be implemented. I searched the site, but I didn't find what I want. Please give me advice.

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    $\begingroup$ Look up the definitions if you don't know them, build the functions. $\endgroup$ – ciao Jul 26 '15 at 3:49

The following antilogarithm function works: antiLog[b_: E, z_] := b^z .

Then the base can be changed via b & since the default for Log is E, antiLog[Log[x]] produces x. However, as GuessWhoItIs pointed out in the comments, Power[~] is worth looking into first. Power[~] is actually exactly equivalent to this.

Similarly, your own cologarithm function can be written such that

coLog[b_: E, z_] := Log[b, 1/z].

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    $\begingroup$ Short version: in Mathematica, the antilogarithm is implemented as Power[]. :) $\endgroup$ – J. M. is away Jul 26 '15 at 3:51
  • $\begingroup$ You forgot to add the base in coLog[]. On the real line, the simpler version is -Log[b, z], but things are different for complex arguments. $\endgroup$ – J. M. is away Jul 26 '15 at 3:53
  • $\begingroup$ Ah thank you, @Guesswhoitis. $\endgroup$ – Ghersic Jul 26 '15 at 3:54
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    $\begingroup$ In light of Guesswhoitis first comment, why not antiLog = Power; ? $\endgroup$ – m_goldberg Jul 26 '15 at 4:29
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    $\begingroup$ @Fernando you may want to accept this answer officially by clicking the grey check mark next to it. $\endgroup$ – MarcoB Aug 5 '15 at 23:16

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