Determine the Base where a Given Equation is True
APL (Dyalog Unicode), 30 bytesSBCS



⊢{3⊃e=×/2↑e←⍵⊥⍺:⍵⋄⍺∇⍵+1}1+⌈/∘,


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Thanks to Adám for the help.

Explanation:

⊢{3⊃e=×/2↑e←⍵⊥⍺:⍵⋄⍺∇⍵+1}1+⌈/∘, ⍝ 
⊢                              ⍝ left argument ⍺: the vector (do nothing)
                        1+⌈/∘, ⍝ right argument ⍵: our starting base.
                             , ⍝             start by flattening the matrix of arguments                               ⌈/   ⍝             reduce by max (find the highest number)
                            ∘  ⍝             compose both of these together
                        1+     ⍝             increment by one
 {         ⍵⊥⍺         }       ⍝ convert inputs to the current base
 {       e←            }       ⍝ store the converted values in 3
 {      2↑             }       ⍝ take the first 2 values
 {    ×/               }       ⍝ multiply them together (reduce-multiply)
 {  e=                 }       ⍝ compare with e (the converted inputs)
 {3⊃                   }       ⍝ only keep the result of the comparison with the 3rd element (expected result)
 {             :⍵      }       ⍝ if truthy, return the current base.
 {               ⋄     }       ⍝ otherwise...
 {                ⍺∇⍵+1}       ⍝ ...recurse with the base incremented


We use a helper function, In, to receive the input into a more palatable format. Otherwise the input is received a matrix of 3 columns.

'3 9 42' would give, for example (read top-down then left-to-right):

0 0 4
3 9 2


And for 'aA bB 36jk' (same here. a is 10, b is 11, A is 36, etc)

 0  0  3
 0  0  6
10 11 19
36 37 20