24^y=50

Yes that is correct but most calculators can't do log base 24.     

$$24^y=50\\\\ log 24^y=log50\\\\ ylog24=log50\\\\ y=\frac{log50}{log24}$$

This log can be any base so long as they are both the same base.

most calcs only do base 10 or base e  

Either of these are fine :)

Using the change of base law your answer can be changed very easily as well

$$log_{b}(a)=\frac{log_n(a)}{log_n(b)}$$

it is  $${{log}}_{{\mathtt{24}}}{\left({\mathtt{50}}\right)}$$  so it is 1.230949