Custom Number Base Converter

MATL, 2 bytes

Za

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All test cases.

For the Za lord!

R, 124 bytes

function(n,s,t,T=L(t),N=(match(!n,!s)-1)%*%L(s)^(L(n):1-1))intToUtf8((!t)[N%/%T^rev(0:log(N,T))%%T+1])
"!"=utf8ToInt
L=nchar

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Ugh, this was a doozy. I use the typical base conversion tricks for R, but string manipulations in R are still messy!

APL (Dyalog Unicode), 22 bytes

Anonymous infix lambda. Takes InputFormat as left argument and OutputFormat as right argument, and prompts for Number from stdin. Assumes ⎕IO (Index Origin) to be 0, which is default on many systems.

{⍵[(≢⍵)⊥⍣¯1⊢(≢⍺)⊥⍺⍳⎕]}

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{…} "dfn"; ⍺ is left argument, ⍵ is right argument
 ^{(mnemonic: left and right ends of the Greek alphabet)}

 ⍵[…] index the output format with the following:

  ⎕ prompt for input

  ⍺⍳ ɩndices of those characters in the input format

  (…)⊥ evaluate as being in the following base:

   ≢⍺ the length of the input format

  ⊢ yield that (separates ¯1 from (≢⍺))

  (…)⊥⍣¯1 convert to the following base:

  ≢⍺ the length of the output format